AMC 10 Statistics and Probability Problems- Year wise

American Mathematics contest 10 (AMC 10) - Statistics problems

AMC 10A 2019 Problem 20

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

$\textbf{(A) }\frac{1}{21}\qquad\textbf{(B) }\frac{1}{14}\qquad\textbf{(C) }\frac{5}{63}\qquad\textbf{(D) }\frac{2}{21}\qquad\textbf{(E) }\frac{1}{7}$

AMC 10A 2019 Problem 22

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \frac{1}{2}$?

$\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{7}{16} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{9}{16} \qquad \textbf{(E) } \frac{2}{3}$

AMC 10A 2018 Problem 11

When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\frac{n}{6^{7}}$, where $n$ is a positive integer. What is $n$?

$\textbf{(A) }42\qquad \textbf{(B) }49\qquad \textbf{(C) }56\qquad \textbf{(D) }63\qquad \textbf{(E) }84\qquad$

AMC 10A 2018 Problem 19

A number $m$ is randomly selected from the set ${11,13,15,17,19}$, and a number $n$ is randomly selected from ${1999,2000,2001,\ldots,2018}$. What is the probability that $m^n$ has a units digit of $1$?

$\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5}$

AMC 10A 2020 Problem 11

What is the median of the following list of $4040$ numbers$?$

$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$

$\textbf{(A)}\ 1974.5\qquad\textbf{(B)}\ 1975.5\qquad\textbf{(C)}\ 1976.5\qquad\textbf{(D)}\ 1977.5\qquad\textbf{(E)}\ 1978.5$

AMC 10A 2020 Problem 2

The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?

$\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$

AMC 10A 2020 Problem 13

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$

$\textbf{(A)}\ \frac12\qquad\textbf{(B)}\ \frac 58\qquad\textbf{(C)}\ \frac 23\qquad\textbf{(D)}\ \frac34\qquad\textbf{(E)}\ \frac 78$

AMC 10A 2020 Problem 15

A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$

AMC 10A 2020 Problem 25

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

AMC 10A 2019 Problem 4

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$

$\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

AMC 10A 2019 Problem 14

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

AMC 10A 2019 Problem 17

A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$

AMC 10B, 2019 Problem 7

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n$?

$\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28$

AMC 10B 2019 Problem 9

The function $f$ is defined by $[f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|]$ for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$?

$\textbf{(A) }$ ${-1, 0}$

$\textbf{(B) }$ $\text{The set of nonpositive integers}$

$\textbf{(C) }$ ${-1, 0, 1}$

$\textbf{(D) }$ ${0}$

$\textbf{(E) }$ $\text{The set of nonnegative integers}$

AMC 10B 2019 Problem 13

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?

$\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}$

AMC 10B 2019 Problem 17

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

$\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$

AMC 10B 2019 Problem 19

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$

$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

AMC 10B, 2019 Problem 21

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

$\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$

AMC 10A 2018 Problem 4

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

AMC 10A 2018 Problem 5

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?

$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$

AMC 10A 2018 Problem 11

When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as[\frac{n}{6^{7}},]where $n$ is a positive integer. What is $n$?

$\textbf{(A) }42\qquad \textbf{(B) }49\qquad \textbf{(C) }56\qquad \textbf{(D) }63\qquad \textbf{(E) }84\qquad$

AMC 10A 2018 Problem 17

Let $S$ be a set of 6 integers taken from ${1,2,\dots,12}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S?$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

AMC 10A 2018 Problem 19

$\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5}$

AMC 10B 2018 Problem 5

How many subsets of ${2,3,4,5,6,7,8,9}$ contain at least one prime number?

$\textbf{(A) }128 \qquad \textbf{(B) }192 \qquad \textbf{(C) }224 \qquad \textbf{(D) }240 \qquad \textbf{(E) }256 \qquad$

AMC 10B 2018 Problem 6

A box contains $5$ chips, numbered $1, 2, 3, 4,$ and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?

$\textbf{(A) }\frac{1}{15} \qquad \textbf{(B) }\frac{1}{10} \qquad \textbf{(C) }\frac{1}{6} \qquad \textbf{(D) }\frac{1}{5} \qquad \textbf{(E) }\frac{1}{4} \qquad$

AMC 10B 2018 Problem 9

The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability $p$?

$\textbf{(A) }13 \qquad \textbf{(B) }26 \qquad \textbf{(C) }32 \qquad \textbf{(D) }39 \qquad \textbf{(E) }42 \qquad$

AMC 10B 2018 Problem 14

A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?

$\textbf{(A) }202 \qquad \textbf{(B) }223 \qquad \textbf{(C) }224 \qquad \textbf{(D) }225 \qquad \textbf{(E) }234 \qquad$

AMC 10B 2018 Problem 18

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?

$\textbf{(A) }60 \qquad \textbf{(B) }72 \qquad \textbf{(C) }92 \qquad \textbf{(D) }96 \qquad \textbf{(E) }120 \qquad$

AMC 10B 2018 Problem 22

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?

$\textbf{(A) }0.21 \qquad \textbf{(B) }0.25 \qquad \textbf{(C) }0.29 \qquad \textbf{(D) }0.50 \qquad \textbf{(E) }0.79 \qquad$

AMC 10A 2017 Problem 12

Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$

$\textbf{(A)}\ \text{a single point}$ $\qquad\textbf{(B)}\ \text{two intersecting lines}$ $\qquad\textbf{(C)}\ \text{ three lines whose pairwise intersections are three distinct points}$ $\qquad\textbf{(D)}\ \text{a triangle}$ $\qquad\textbf{(E)}\ \text{three rays with a common endpoint}$

AMC 10A 2017 Problem 15

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0, 4034]$. What is the probability that Laurent's number is greater than Chloé's number? (Assume they cannot be equal)

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{6}\qquad\textbf{(E)}\ \frac{7}{8}$

AMC 10A 2017 Problem 18

Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

AMC 10A 2017 Problem 19

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?

$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

AMC 10A 2017 Problem 9

A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?

$\textbf{(A)}\ \frac{1}{27}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{7}{27}\qquad\textbf{(E)}\ \frac{1}{2}$

AMC 10A 2017 Problem 25

How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.

$\textbf{(A)}\ 226\qquad\textbf{(B)}\ 243\qquad\textbf{(C)}\ 270\qquad\textbf{(D)}\ 469\qquad\textbf{(E)}\ 486$

AMC 10B 2017 Problem 14

An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?

$\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1$

AMC 10B 2017 Problem 18

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$

AMC 10B 2017 Problem 20

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

$\textbf{(A)}\ \frac{1}{21} \qquad \textbf{(B)}\ \frac{1}{19} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{11}{21}$

AMC 10A 2016 Problem 7

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$?

$\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$

AMC 10A 2016 Problem 12

Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?

$\textbf{(A)}\ p<\frac{1}{8}\qquad\textbf{(B)}\ p=\frac{1}{8}\qquad\textbf{(C)}\ \frac{1}{8}<p<\frac{1}{3}\qquad\textbf{(D)}\ p=\frac{1}{3}\qquad\textbf{(E)}\ p>\frac{1}{3}$

AMC 10A 2016 Problem 14

How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.)

$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

AMC 10A 2016 Problem 17

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \frac{321}{400}$?

$\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

AMC 10A 2016 Problem 18

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?

$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

AMC 10B 2016 Problem 6

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

AMC 10B 2016 Problem 12

Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even?

$\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

AMC 10B 2016 Problem 16

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$

$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

AMC 10B 2016 Problem 17

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

$\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680$

AMC 10B 2016 Problem 22

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams ${A, B, C}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$

$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

AMC 10A 2015 Problem 10

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$.

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

AMC 10A 2015 Problem 25

Let $S$ be a square of side length $1$. Two points are chosen at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers with $\gcd(a,b,c)=1$. What is $a+b+c$?

$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

AMC 10B 2015 Problem 11

How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.)

$\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

AMC 10B 2015 Problem 15

The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

$\textbf{(A) } 41 \qquad\textbf{(B) } 47 \qquad\textbf{(C) } 59 \qquad\textbf{(D) } 61 \qquad\textbf{(E) } 66$

AMC 10B 2015 Problem 16

Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?

$\textbf{(A) } \frac{9}{1000} \qquad\textbf{(B) } \frac{1}{90} \qquad\textbf{(C) } \frac{1}{80} \qquad\textbf{(D) } \frac{1}{72} \qquad\textbf{(E) } \frac{2}{121}$

AMC 10B 2015 Problem 18

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?

$\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64$

AMC 10A 2014 Problem 4

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

AMC 10A 2014 Problem 17

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

$\textbf{(A)}\ \frac16\qquad\textbf{(B)}\ \frac{13}{72}\qquad\textbf{(C)}\ \frac7{36}\qquad\textbf{(D)}\ \frac5{24}\qquad\textbf{(E)}\ \frac29$

AMC 10B 2014 Problem 16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

$\textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9}\qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}$

AMC 10B 2014 Problem 18

A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?

$\textbf {(A) } 24 \qquad \textbf {(B) } 30 \qquad \textbf {(C) } 31\qquad \textbf {(D) } 33 \qquad \textbf {(E) } 35$

AMC 10B 2014 Problem 24

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

$\textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5$

AMC 10B 2014 Problem 25

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?

$\textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2}$

AMC 10A 2013 Problem 7

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$

AMC 10A 2013 Problem 11

A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$

AMC 10A 2013 Problem 24

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

$\textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900$

AMC 10B 2013 Problem 12

Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?

$\textbf{(A) }\frac{2}5\qquad\textbf{(B) }\frac{4}9\qquad\textbf{(C) }\frac{1}2\qquad\textbf{(D) }\frac{5}9\qquad\textbf{(E) }\frac{4}5$

AMC 10A 2012 Problem 9

A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$

AMC 10A 2012 Problem 20

A $3$ x $3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?

$\textbf{(A)}\ \frac{49}{512}\qquad\textbf{(B)}\ \frac{7}{64}\qquad\textbf{(C)}\ \frac{121}{1024}\qquad\textbf{(D)}\ \frac{81}{512}\qquad\textbf{(E)}\ \frac{9}{32}$

AMC 10A 2012 Problem 23

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

$\textbf{(A)}\ 60\qquad\textbf{(B)}\ 170\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 320\qquad\textbf{(E)}\ 660$

AMC 10A 2012 Problem 25

Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

AMC 10B 2012 Problem 11

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

$\textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304$

AMC 10B 2012 Problem 15

In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

AMC 10B 2012 Problem 24

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?

$\textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105$

AMC 10A 2011 Problem 6

Set $A$ has 20 elements, and set $B$ has 15 elements. What is the smallest possible number of elements in $A \cup B$, the union of $A$ and $B$?

$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 35\qquad\textbf{(E)}\ 300$

AMC 10A 2011 Problem 14

A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

$\textbf{(A)}\,\frac{1}{36} \qquad\textbf{(B)}\,\frac{1}{12} \qquad\textbf{(C)}\,\frac{1}{6} \qquad\textbf{(D)}\,\frac{1}{4} \qquad\textbf{(E)}\,\frac{5}{18}$

AMC 10A 2011 Problem 20

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$

AMC 10A 2011 Problem 21

Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?

$\textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16}$

AMC 10B 2011 Problem 13

Two real numbers are selected independently at random from the interval $[-20, 10]$. What is the probability that the product of those numbers is greater than zero?

$\textbf{(A)}\ \frac{1}{9} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{4}{9} \qquad\textbf{(D)}\ \frac{5}{9} \qquad\textbf{(E)}\ \frac{2}{3}$

AMC 10B 2011 Problem 16

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?

$\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

AMC 10A 2010 Problem 18

Bernardo randomly picks 3 distinct numbers from the set ${1,2,3,4,5,6,7,8,9}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set ${1,2,3,4,5,6,7,8}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

$\textbf{(A)}\ \frac{47}{72} \qquad \textbf{(B)}\ \frac{37}{56} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{49}{72} \qquad \textbf{(E)}\ \frac{39}{56}$

AMC 10A 2010 Problem 23

Each of 2010 boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?

$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

AMC 10A 2010 Problem 25

\begin{array}{ccccc} {}&{}&{}&{}&55\ 55&-&7^2&=&6\ 6&-&2^2&=&2\ 2&-&1^2&=&1\ 1&-&1^2&=&0\ \end{array}
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$?

$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 3 \qquad \mathrm{(C)}\ 5 \qquad \mathrm{(D)}\ 7 \qquad \mathrm{(E)}\ 9$

AMC 10B 2010 Problem 3

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
[/et_pb_team_member][et_pb_team_member name=

AMC 10B 2010 Problem 17

Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?

$\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26$

AMC 10B 2010 Problem 18

Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set ${1, 2, 3,\dots, 2010}$. What is the probability that $abc + ab + a$ is divisible by $3$?

$\textbf{(A)}\ \frac{1}{3} \qquad \textbf{(B)}\ \frac{29}{81} \qquad \textbf{(C)}\ \frac{31}{81} \qquad \textbf{(D)}\ \frac{11}{27} \qquad \textbf{(E)}\ \frac{13}{27}$

AMC 10B 2010 Problem 21

A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?

$\textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{7} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{5}$

AMC 10B 2010 Problem 22

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?

$\textbf{(A)}\ 1930 \qquad \textbf{(B)}\ 1931 \qquad \textbf{(C)}\ 1932 \qquad \textbf{(D)}\ 1933 \qquad \textbf{(E)}\ 1934$

AMC 10B 2010 Problem 23

The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$

AMC 10 Combinatorics Questions - Year wise

American Mathematics contest 10 (AMC 10) - Combinatorics problems

Try these AMC 10 Combinatorics Questions and check your knowledge

AMC 10A, 2020, Problem 9

A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$

$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$

AMC 10A, 2020, Problem 15

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$

AMC 10A, 2019, Problem 14

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

AMC 10A, 2019, Problem 17

$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$

AMC 10A, 2019, Problem 20

(A) $\frac{1}{21}$ (B) $\frac{1}{4}$ (C) $\frac{5}{63}$ (D) $\frac{2}{21}$ (E) $\frac{1}{7}$

AMC 10A, 2018, Problem 4

(A) $3$ (B) $6$ (C) $12$ (D) $18$ (E) $24$

AMC 10 Algebra Previous Year Questions - Year wise

Get rolling on your preparation for AMC 10 with Cheenta. This post has all the AMC 10 Algebra previous year Questions, year-wise. Try out these problems:

AMC 10A, 2021, Problem 1

What is the value of

$\left(2^{2}-2\right)-\left(3^{2}-3\right)+\left(4^{2}-4\right)$

(A) 1
(B) 2
(C) 5
(D) 8
(E) 12

AMC 10A, 2021, Problem 2

Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 students. How many students does Portia's high school have?
(A) 600
(B) 650
(C) 1950
(D) 2000
(E) 2050

AMC 10A, 2021, Problem 3

The sum of two natural numbers is 17,402 . One of the two numbers is divisible by 10 . If the units digit of that number is erased, the other number is obtained.

What is the difference of these two numbers?
(A) 10,272
(B) 11,700
(C) 13,362
(D) 14,238
(E) 15,426

AMC 10A, 2021, Problem 4

A cart rolls down a hill, travelling 5 inches the first second and accelerating so that during each successive 1 -second time interval, it travels inches more than during the previous 1 -second interval.

The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel?
(A) 215
(B) 360
(C) 2992
(D) 3195
(E) 3242

AMC 10A, 2021, Problem 5

The quiz scores of a class with $k>12$ students have a mean of 8 . The mean of a collection of 12 of these quiz scores is 14 .

What is the mean of the remaining quiz scores in terms of $k$ ?
(A) $\frac{14-8}{k-12}$
(B) $\frac{8 k-168}{k-12}$
(C) $\frac{14}{12}-\frac{8}{k}$
(D) $\frac{14(k-12)}{k^{2}}$
(E) $\frac{14(k-12)}{8 k}$

AMC 10A, 2021, Problem 6

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking a 4 miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 2 miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 3 miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

(A) $\frac{12}{13}$
(B) $1$
(C) $\frac{13}{12}$
(D) $\frac{24}{13}$
(E) $2$

AMC 10B, 2021, Problem 2

What is the value of $\sqrt{(3-2 \sqrt{3})^{2}}+\sqrt{(3+2 \sqrt{3})^{2}}$ ?
(A) 0
(B) $4 \sqrt{3}-6$
(C) 6
(D) $4 \sqrt{3}$
(E) $4 \sqrt{3}+6$

AMC 10B, 2021, Problem 15

The real number $x$ satisfies the equation $x+\frac{1}{x}=\sqrt{5}$.

What is the value of $x^{11}-7 x^{7}+x^{3} ?$
(A) $-1$
(B) $0$
(C) $1$
(D) $2$
(E) $\sqrt{5}$

AMC 10A, 2020, Problem 22

Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2 , the second sheet contains pages 3 and 4 , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 19 . How many sheets were borrowed?
(A) 10
(B) 13
(C) 15
(D) 17
(E) 20

AMC 10A, 2020, Problem 1

What value of $x$ satisfies $x-\frac{3}{4}=\frac{5}{12}-\frac{1}{3} ?$

(A) $-\frac{2}{3}$
(B) $\frac{7}{36}$
(C) $\frac{7}{12}$
(D) $\frac{2}{3}$
(E) $\frac{5}{6}$

AMC 10A, 2020, Problem 2

The numbers $3,5,7, a$, and $b$ have an average (arithmetic mean) of $15$ .

What is the average of $a$ and $b$ ?
(A) 0
(B) 15
(C) 30
(D) 45
(E) 60

AMC 10A, 2020, Problem 3

Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression?

$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$

(A) $-1$
(B) 1
(C) $\frac{a b c}{60}$
(D) $\frac{1}{a b c}-\frac{1}{60}$
(E) $\frac{1}{60}-\frac{1}{a b c}$

AMC 10A, 2020, Problem 5

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

(A) $12$
(B) $15$
(C) $18$
(D) $21$
(E) $25$

AMC 10A, 2020, Problem 14

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$.

What is the value of $x+\frac{x^{3}}{y^{2}}+\frac{y^{3}}{x^{2}}+y ?$

(A) $360$
(B) $400$
(C) $420$
(D) $440$
(E) $480$

AMC 10B, 2020, Problem 1

What is the value of

1 - (- 2) - 3 - (- 4) - 5 - (- 6) ?

$1-(-2)-3-(-4)-5-(-6) ?$

(A) $-20$
(B) $-3$
(C) $3$
(D) $5$
(E) $21$

AMC 10B, 2020, Problem 3

The ratio of $w$ to $x$ is $4: 3$, the ratio of $y$ to $z$ is $3: 2$,

and the ratio of $z$ to $x$ is $1: 6$. What is the ratio of $w$ to $y$ ?
(A) $4: 3$
(B) $3: 2$
(C) $8: 3$
(D) $4: 1$
(E) $16: 3$

AMC 10B, 2020, Problem 6

Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later, the odometer displayed the next higher palindrome.

What was her average speed, in miles per hour, during this 2 hour period?
(A) 50
(B) 55
(C) 60
(D) 65
(E) 70

AMC 10B, 2020, Problem 7

How many positive even multiples of 3 less than 2020 are perfect squares?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 12

AMC 10B, 2020, Problem 9

How many ordered pairs of integers $(x, y)$ satisfy the equation

x 2020 + y 2 = 2 y ?

$x^{2020}+y^{2}=2 y ?$

(A) 1
(B) 2
(C) 3
(D) 4
(E) infinitely many

AMC 10B, 2020, Problem 12

The decimal representation of

1 20 2

$\frac{1}{20^{2}}$

consists of a string of zeros after the decimal point, followed by a 9 and then several more digits.

How many zeros are in that initial string of zeros after the decimal point?
(A) 23
(B) 24
(C) 25
(D) 26
(E) 27

AMC 10B, 2020, Problem 22

What is the remainder when $2^{202}+202$ is divided by $2^{101}+2^{51}+1$ ?
(A) 100
(B) 101
(C) 200
(D) 201
(E) 202

AMC 10A, 2019, Problem 1

What is the value of

${ }_{2}^{\left(0^{\left(1^{9}\right)}\right)}+\left(\left(2^{0}\right)^{1}\right)^{9} ?$

(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$

AMC 10A, 2019, Problem 2

What is the hundreds digit of $(20 !-15 !) ?$
(A) $0$
(B) $1$
(C) $2$
(D) $4$
(E) $5$

AMC 10A, 2019, Problem 3

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$ ?
(A) 3
(B) 5
(C) 9
(D) 12
(E) 15

AMC 10A, 2019, Problem 24

Let $p, q$, and $r$ be the distinct roots of the polynomial $x^{3}-22 x^{2}+80 x-67$. It is given that there exist real numbers $A, B$, and $C$ such that

$\frac{1}{s^{3}-22 s^{2}+80 s-67}=\frac{A}{s-p}+\frac{B}{s-q}+\frac{C}{s-r}$

for all $s \notin{p, q, r}$. What is $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$ ?

(A) $243$
(B) $244$
(C) $245$
(D) $246$
(E) $247$

AMC 10B, 2019, Problem 1

Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
(A) $\frac{5}{8}$
(B) $\frac{4}{5}$
(C) $\frac{7}{8}$
(D) $\frac{9}{10}$
(E) $\frac{11}{12}$

AMC 10B, 2019, Problem 2

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
(A) 11
(B) 15
(C) 19
(D) 21
(E) 27

AMC 10B, 2019, Problem 3

In a high school with 500 students, $40 \%$ of the seniors play a musical instrument, while $30 \%$ of the non-seniors do not play a musical instrument. In all, $46.8 \%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
(A) 66
(B) 154
(C) 186
(D) 220
(E) 266

AMC 10B, 2019, Problem 4

All lines with equation $a x+b y=c$ such that $a, b, c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?
(A) $(-1,2)$
(B) $(0,1)$
(C) $(1,-2)$
(D) $(1,0)$
(E) $(1,2)$

AMC 10B, 2019, Problem 11

Two jars each contain the same number of marbles, and every marble is either blue or green.

In Jar 1 the ratio of blue to green marbles is $9: 1$, and the ratio of blue to green marbles in Jar 2 is $8: 1$. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar $2 ?$
(A) 5
(B) 10
(C) 25
(D) 45
(E) 50

AMC 10B, 2019, Problem 13

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17$, and $x$ is equal to the mean of those five numbers?
(A) $-5$
(B) 0
(C) 5
(D) $\frac{15}{4}$
(E) $\frac{35}{4}$

AMC 10B, 2019, Problem 18

Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$ ?
(A) $\frac{2}{3}$
(B) 1
(C) $\frac{6}{5}$
(D) $\frac{5}{4}$
(E) $\frac{3}{2}$

AMC 10A, 2018, Problem 1

What is the value of

(((2 + 1) - 1 + 1) - 1 + 1) - 1 + 1 ?

$\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 ?$

(A) $\frac{5}{8}$
(B) $\frac{11}{7}$
(C) $\frac{8}{5}$
(D) $\frac{18}{11}$
(E) $\frac{15}{8}$

AMC 10A, 2018, Problem 2

Liliane has $50 \%$ more soda than Jacqueline, and Alice has $25 \%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
(A) Liliane has $20 \%$ more soda than Alice.
(B) Liliane has $25 \%$ more soda than Alice.
(C) Liliane has $45 \%$ more soda than Alice.
(D) Liliane has $75 \%$ more soda than Alice.
(E) Liliane has $100 \%$ more soda than Alice.

AMC 10A, 2018, Problem 3

A unit of blood expires after $10 !=10 \cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1 .$ On what day does his unit of blood expire?
(A) January 2
(B) January 12
(C) January 22
(D) February 11
(E) February 12

AMC 10A, 2018, Problem 6

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0 , and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90 , and that $65 \%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
(A) 200
(B) 300
(C) 400
(D) 500
(E) 600

AMC 10A, 2018, Problem 8

Joe has a collection of 23 coins, consisting of 5 -cent coins, 10 -cent coins, and 25 -cent coins. He has 3 more 10 -cent coins than 5 -cent coins and the total value of his collection is 320 cents. How many more 25 -cent coins does Joe have than 5 -cent coins?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

AMC 10A, 2018, Problem 10

Suppose that real number $x$ satisfies

49 - x 2 - - - - - - \sqrt - 25 - x 2 - - - - - - \sqrt = 3

$\sqrt{49-x^{2}}-\sqrt{25-x^{2}}=3$

What is the value of $\sqrt{49-x^{2}}+\sqrt{25-x^{2}}$ ?
(A) 8
(B) $\sqrt{33}+8$
(C) 9
(D) $2 \sqrt{10}+4$
(E) 12

AMC 10B, 2018, Problem 1

Kate bakes a 20-inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
(A) 90
(B) 100
(C) 180
(D) 200
(E) 360

AMC 10B, 2018, Problem 2

Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was $60 \mathrm{mph}$ (miles per hour), and his average speed during the second 30 minutes was $65 \mathrm{mph}$. What was his average speed, in mph, during the last 30 minutes?
(A) 64
(B) 65
(C) 66
(D) 67
(E) 68

AMC 10A, 2018, Problem 4

A three-dimensional rectangular box with dimensions $X, Y$, and $Z$ has faces whose surface areas are $24,24,48,48,72$, and 72 square units. What is $X+Y+Z ?$
(A) 18
(B) 22
(C) 24
(D) 30
(E) 36

AMC 10A, 2018, Problem 19

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

AMC 10A, 2017, Problem 1

What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$ ?
(A) 70
(B) 97
(C) 127
(D) 159
(E) 729

AMC 10A, 2017, Problem 2

Pablo buys popsicles for his friends. The store sells single popsicles for $\$ 1$ each, 3-popsicle boxes for $\$ 2$ each, and 5-popsicle boxes for $\$ 3$ What is the greatest number of popsicles that Pablo can buy with $\$ 8$ ?
(A) $8$
(B) $11$
(C) $12$
(D) $13$
(E) $15$

AMC 10A, 2017, Problem 4

Mia is "helping" her mom pick up 30 toys that are strewn on the floor. Mia's mom manages to put 3 toys into the toy box every 30 seconds, but each time immediately after those 30 seconds have elapsed, Mia takes 2 toys out of the box. How much time, in minutes, will it take Mia and her mom to put all 30 toys into the box for the first time?
(A) $13.5$
(B) 14
(C) $14.5$
(D) 15
(E) $15.5$

AMC 10A, 2017, Problem 5

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
(A) 1
(B) 2
(C) 4
(D) 8
(E) 12

AMC 10A, 2017, Problem 9

Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at $30 \mathrm{kph}$, and uphill at $5 \mathrm{kph}$. Penny rides on a flat road at $30 \mathrm{kph}$, downhill at $40 \mathrm{kph}$, and uphill at $10 \mathrm{kph}$. Minnie goes from town $A$ to town $B$, a distance of $10 \mathrm{~km}$ all uphill, then from town $B$ to town $C$, a distance of 15 $\mathrm{km}$ all downhill, and then back to town $A$, a distance of $20 \mathrm{~km}$ on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the 45 -km ride than it takes Penny?
(A) 45
(B) 60
(C) 65
(D) 90
(E) 95

AMC 10A, 2017, Problem 10

Joy has 30 thin rods, one each of every integer length from $1 \mathrm{~cm}$ through $30 \mathrm{~cm}$. She places the rods with lengths $3 \mathrm{~cm}, 7 \mathrm{~cm}$, and $15 \mathrm{~cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
(A) 16
(B) 17
(C) 18
(D) 19
(E) 20

AMC 10A, 2017, Problem 14

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20 \%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5 \%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
(A) $9 \%$
(B) $19 \%$
(C) $22 \%$
(D) $23 \%$
(E) $25 \%$

AMC 10A, 2017, Problem 16

There are 10 horses, named Horse 1, Horse $2, \ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S>0$, in minutes, at which all 10 horses will again simultaneously be at the starting point is $S=2520$. Let $T>0$ be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of $T$ ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

AMC 10B, 2017, Problem 1

Mary thought of a positive two-digit number. She multiplied it by 3 and added 11 . Then she switched the digits of the result, obtaining a number between 71 and 75 , inclusive. What was Mary's number?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

AMC 10B, 2017, Problem 2

Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?
(A) 5 minutes and 35 seconds
(B) 6 minutes and 40 seconds
(C) 7 minutes and 5 seconds
(D) 7 minutes and 25 seconds
(E) 8 minutes and 10 seconds

AMC 10B, 2017, Problem 3

Real numbers $x, y$, and $z$ satisfy the inequalities $0<x<1,-1<y<0$, and $1<z<2$. Which of the following numbers is necessarily positive?
(A) $y+x^{2}$
(B) $y+x z$
(C) $y+y^{2}$
(D) $y+2 y^{2}$
(E) $y+z$

AMC 10B, 2017, Problem 4

Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3 x+y}{x-3 y}=-2$. What is the value of $\frac{x+3 y}{3 x-y}$ ?
(A) $-3$
(B) $-1$
(C) 1
(D) 2
(E) 3

AMC 10B, 2017, Problem 5

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

AMC 10B, 2017, Problem 7

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
(A) $2.0$
(B) $2.2$
(C) $2.8$
(D) $3.4$
(E) $4.4$

AMC 10B, 2017, Problem 10

The lines with equations $a x-2 y=c$ and $2 x+b y=-c$ are perpendicular and intersect at $(1,-5)$. What is $c ?$
$(\mathbf{A})-13$
(B) $-8$
(C) 2
(D) 8
(E) 13

AMC 10B, 2017, Problem 11

At Typico High School, $60 \%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80 \%$ say that they like it, and the res say that they dislike it. Of those who dislike dancing, $90 \%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
(A) $10 \%$
(B) $12 \%$
(C) $20 \%$
(D) $25 \%$
(E) $33 \frac{1}{3} \%$

AMC 10B, 2017, Problem 12

Elmer's new car gives $50 \%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel which is $20 \%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
(A) $20 \%$
(B) $26 \frac{2}{3} \%$
(C) $27 \frac{7}{9} \%$
(D) $33 \frac{1}{3} \%$
(E) $66 \frac{2}{3} \%$

AMC 10B, 2017, Problem 13

There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2016, Problem 1

What is the value of $\frac{11 !-10 !}{9 !}$ ?
(A) $99$
(B) $100$
(C) $110$
(D) $121$
(E) $132$

AMC 10A, 2016, Problem 2

For what value of $x$ does $10^{x} \cdot 100^{2 x}=1000^{5}$ ?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2016, Problem 3

For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $\$ 12.50$ more than David. How much did they spend in the bagel store together?
(A) $\$ 37.50$
(B) $\$ 50.00$
(C) $\$ 87.50$
(D) $\$ 90.00$
(E) $\$ 92.50$

AMC 10A, 2016, Problem 5

A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
(A) $48$
(B) $56$
(C) $64$
(D) $96$
(E) $144$

AMC 10A, 2016, Problem 6

Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit 1$$ Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
(A) $13$
(B) $26$
(C) $102$
(D) $103$
(E) $110$

AMC 10A, 2016, Problem 7

The mean, median, and mode of the $7$ data values $60,100, x, 40,50,200,90$ are all equal to $x$. What is the value of $x$ ?
(A) $50$
(B) $60$
(C) $75$
(D) $90$
(E) $100$

AMC 10A, 2016, Problem 8

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
(A) $20$
(B) $30$
(C) $35$
(D) $40$
(E) $45$

AMC 10B, 2016, Problem 1

What is the value of $\frac{2 a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a=\frac{1}{2}$ ?
(A) $1$
(B) $2$
(C) $\frac{5}{2}$
(D) $10$
(E) $20$

AMC 10B, 2016, Problem 3

Let $x=-2016$. What is the value of ||$|x|-x|-| x||-x$ ?
(A) $-2016$
(B) 0
(C) 2016
(D) 4032
(E) 6048

AMC 10B, 2016, Problem 5

The mean age of Amanda's $4$ cousins is $8$ , and their median age is $5$ . What is the sum of the ages of Amanda's youngest and oldest cousins?
(A) $13$
(B) $16$
(C) $19$
(D) $22$
(E) $25$

AMC 10B, 2016, Problem 6

AMC 10B, 2016, Problem 7

The ratio of the measures of two acute angles is $5: 4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
(A) $75$
(B) $90$
(C) $135$
(D) $150$
(E) $270$

AMC 10B, 2016, Problem 8

What is the tens digit of $2015^{2016}-2017 ?$
(A) $0$
(B) $1$
(C) $3$
(D) $5$
(E) $8$

AMC 10B, 2016, Problem 10

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
(A) $14.0$
(B) $16.0$
(C) $20.0$
(D) $33.3$
(E) $55.6$

AMC 10B, 2016, Problem 13

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets o triplets. How many of these 1000 babies were in sets of quadruplets?
(A) $25$
(B) $40$
(C) $64$
(D) $100$
(E) $160$

AMC 10B, 2016, Problem 16

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is 1 . What is the smallest possible value of $S$ ?
(A) $\frac{1+\sqrt{5}}{2}$
(B) $2$
(C) $\sqrt{5}$
(D) $3$
(E) $4$

AMC 10A, 2015, Problem 1

What is the value of $\left(2^{0}-1+5^{2}-0\right)^{-1} \times 5 ?$
(A) $-125$
(B) $-120$
(C) $\frac{1}{5}$
(D) $\frac{5}{24}$
(E) $25$

AMC 10A, 2015, Problem 2

A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?
(A) $3$
(B) $5$
(C) $7$
(D) $9$
(E) $11$

AMC 10A, 2015, Problem 4

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$

AMC 10A, 2015, Problem 5

Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$ . After he graded Payton's test, the test average became $81$ . What was Payton's score on the test?
(A) $81$
(B) $85$
(C) $91$
(D) $94$
(E) $95$

AMC 10A, 2015, Problem 6

The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller number?
(A) $\frac{5}{4}$
(B) $\frac{3}{2}$
(C) $\frac{9}{5}$
(D) $2$
(E) $\frac{5}{2}$

AMC 10A, 2015, Problem 7

How many terms are in the arithmetic sequence $13,16,19, \ldots, 70,73$ ?
(A) $20$
(B) $21$
(C) $24$
(D) $60$
(E) $61$

AMC 10A, 2015, Problem 8

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2: 1$ ?
(A) $2$
(B) $4$
(C) $5$
(D) $6$
(E) $8$

AMC 10A, 2015, Problem 11

The ratio of the length to the width of a rectangle is $4: 3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $k d^{2}$ for some constant $k$. What is $k$ ?
(A) $\frac{2}{7}$
(B) $\frac{3}{7}$
(C) $\frac{12}{25}$
(D) $\frac{16}{25}$
(E) $\frac{3}{4}$

AMC 10A, 2015, Problem 12

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^{2}+x^{4}=2 x^{2} y+1$. What is $|a-b|$ ?
(A) $1$
(B) $\frac{\pi}{2}$
(C) $2$
(D) $\sqrt{1+\pi}$
(E) $1+\sqrt{\pi}$

AMC 10A, 2015, Problem 15

Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10 \%$ ?
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) infinitely many

AMC 10A, 2015, Problem 16

If $y+4=(x-2)^{2}, x+4=(y-2)^{2}$, and $x \neq y$, what is the value of $x^{2}+y^{2}$ ?
(A) $10$
(B) $15$
(C) $20$
(D) $25$
(E) $30$

AMC 10B, 2015, Problem 1

What is the value of $2-(-2)^{-2}$ ?
(A) $-2$
(B) $\frac{1}{16}$
(C) $\frac{7}{4}$
(D) $\frac{9}{4}$
(E) $6$

AMC 10B, 2015, Problem 2

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1: 00 \mathrm{PM}$ and finishes the seconc task at $2: 40 \mathrm{PM}$. When does she finish the third task?
(A) $3:10$ PM
(B) $3:30$ PM
(C) $4:00$ PM
(D) $4:10$ PM
(E) $4:30$ PM

AMC 10B, 2015, Problem 3

Kaashish has written down one integer two times and another integer three times. The sum of the five numbers is $100$ , and one of the numbers is $28$ . What is the other number?
(A) $8$
(B) $11$
(C) $14$
(D) $15$
(E) $18$

AMC 10B, 2015, Problem 7

Consider the operation "minus the reciprocal of," defined by $a \diamond b=a-\frac{1}{b}$. What is $((1 \diamond 2) \diamond 3)-(1 \diamond(2 \diamond 3))$ ?

(A) $-\frac{7}{30}$
(B) $-\frac{1}{6}$
(C) $0$
(D) $\frac{1}{6}$
(E) $\frac{7}{30}$

AMC 10B, 2015, Problem 13

The line $12 x+5 y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
(A) $20$
(B) $\frac{360}{17}$
(C) $\frac{107}{5}$
(D) $\frac{43}{2}$
(E) $\frac{281}{13}$

AMC 10B, 2015, Problem 14

Let $a, b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0 ?$
(A) $15$
(B) $15.5$
(C) $16$
(D) $16.5$
(E) $17$

AMC 10A, 2014, Problem 1

What is $10 \cdot\left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}$ ?
(A) $3$
(B) $8$
(C) $\frac{25}{2}$
(D) $\frac{170}{3}$
(E) $170$

AMC 10A, 2014, Problem 3

Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\$ 2.50$ each. In the afternoon she sells two thirds o what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$ 0.75$ for her to make. In dollars, what is her profit for the day?
(A) $24$
(B) $36$
(C) $44$
(D) $48$
(E) $52$

AMC 10A, 2014, Problem 5

On an algebra quiz, $10 \%$ of the students scored 70 points, $35 \%$ scored 80 points, $30 \%$ scored 90 points, and the rest scored 100 points. What is the difference between the mean and median score of the students' scores on this quiz?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2014, Problem 7

Nonzero real numbers $x, y, a$, and $b$ satisfy $x<a$ and $y<b$. How many of the following inequalities must be true?
(I) $x+y<a+b$
(II) $x-y<a-b$
(III) $x y<a b$
(IV) $\frac{x}{y}<\frac{a}{b}$
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$

AMC 10A, 2014, Problem 8

Which of the following numbers is a perfect square?
(A) $\frac{14 ! 15 !}{2}$
(B) $\frac{15 ! 16 !}{2}$
(C) $\frac{16 ! 17 !}{2}$
(D) $\frac{17 ! 18 !}{2}$
(E) $\frac{18 ! 19 !}{2}$

AMC 10A, 2014, Problem 10

Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$ ?
(A) $a+3$
(B) $a+4$
(C) $a+5$
(D) $a+6$
(E) $a+7$

AMC 10A, 2014, Problem 11

A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: $10 \%$ off the listed price if the listed price is at least $\$ 50$
Coupon 2: $\$ 20$ off the listed price if the listed price is at least $\$ 100$
Coupon 3: $18 \%$ off the amount by which the listed price exceeds $\$ 100$
For which of the following listed prices will coupon 1 offer a greater price reduction than either coupon $2$ or coupon $3$ ?
(A) $\$ 179.95$
(B) $\$ 199.95$
(C) $\$ 219.95$
(D) $\$ 239.95$
$(\mathbf{E}) \$ 259.95$

AMC 10A, 2014, Problem 15

David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?
(A) $140$
(B) $175$
(C) $210$
(D) $245$
(E) $280$

AMC 10B, 2014, Problem 1

Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
(A) $33$
(B) $35$
(C) $37$
(D) $39$
(E) $41$

AMC 10B, 2014, Problem 2

What is $\frac{2^{3}+2^{3}}{2^{-3}+2^{-3} ?} ?$
(A) $16$
(B) $24$
(C) $32$
(D) $48$
(E) $64$

AMC 10B, 2014, Problem 3

Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip?
(A) $30$
(B) $\frac{400}{11}$
(C) $\frac{75}{2}$
(D) $40$
(E) $\frac{300}{7}$

AMC 10B, 2014, Problem 4

Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
(A) $\frac{3}{2}$
(B) $\frac{5}{3}$
(C) $\frac{7}{4}$
(D) $2$
(E) $\frac{13}{4}$

AMC 10B, 2014, Problem 6

Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons:
buy 1 balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
(A) $33$
(B) $34$
(C) $36$
(D) $38$
(E) $39$

AMC 10B, 2014, Problem 7

Suppose $A>B>0$ and $\mathrm{A}$ is $x \%$ greater than $B$. What is $x$ ?
(A) $100\left(\frac{A-B}{B}\right)$
(B) $100\left(\frac{A+B}{B}\right)$
(C) $100\left(\frac{A+B}{A}\right)$
(D) $100\left(\frac{A-B}{A}\right)$
(E) $100\left(\frac{A}{B}\right)$

AMC 10B, 2014, Problem 8

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?
(A) $\frac{b}{1080 t}$
(B) $\frac{30 t}{b}$
(C) $\frac{30 b}{t}$
(D) $\frac{10 t}{b}$
(E) $\frac{10 b}{t}$

AMC 10B, 2014, Problem 9

For real numbers $w$ and $z$,

1 w + 1 z 1 w - 1 z = 2014

$\frac{\frac{1}{w}+\frac{1}{z}}{\frac{1}{w}-\frac{1}{z}}=2014$

What is $\frac{w+z}{w-z} ?$
(A) $-2014$
(B) $\frac{-1}{2014}$
(C) $\frac{1}{2014}$
(D) $1$
(E) $2014$

AMC 10A, 2013, Problem 1

A taxi ride costs $\$ 1.50$ plus $\$ 0.25$ per mile traveled. How much does a 5 -mile taxi ride cost?
(A) $2.25$
(B) $2.50$
(C) $2.75$
(D) $3.00$
(E) $3.75$

AMC 10A, 2013, Problem 2

Alice is making a batch of cookies and needs $2 \frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
(A) $8$
(B) $10$
(C) $12$
(D) $16$
(E) $20$

AMC 10A, 2013, Problem 5

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$ 105$, Dorothy paid $\$ 125$, and Sammy paid $\$ 175$. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$ ?
(A) $15$
(B) $20$
(C) $25$
(D) $30$
(E) $35$

AMC 10A, 2013, Problem 6

Joey and his five brothers are ages $3,5,7,9,11$, and 13 . One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5 -year-old stayed home. How old is Joey?
(A) $3$
(B) $7$
(C) $9$
(D) $11$
(E) $13$

AMC 10A, 2013, Problem 8

What is the value of

2 2014 + 2 2012 2 2014 - 2 2012 ?

$\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?$

(A) $-1$
(B) $1$
(C) $\frac{5}{3}$
(D) $2013$
$(\mathbf{E}) 2^{4024}$

AMC 10A, 2013, Problem 9

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20 \%$ of her three-point shots and $30 \%$ of her two-point shots. Shenille attempted 30 shots. How many points did she score?
(A) $12$
(B) $18$
(C) $24$
(D) $30$
(E) $36$

AMC 10A, 2013, Problem 10

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

(A) $15$

(B) $30$

(D) $70$

(E) $60$

AMC 10A, 2013, Problem 14

A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$ . How many edges does the remaining solid have?
(A) $36$
(B) $60$
(C) $72$
(D) $84$
(E) $108$

AMC 10B, 2013, Problem 1

What is $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} ?$

(A) $\frac{7}{12}$

(B) $\frac{49}{20}$

(D) $\frac{5}{36}$

(E) $-1$

AMC 10B, 2013, Problem 2

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr. Green's steps is 2 feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
(A) $600$
(B) $800$
(C) $1000$
(D) $1200$
(E) $1400$

AMC 10B, 2013, Problem 10

A basketball team's players were successful on $50 \%$ of their two-point shots and $40 \%$ of their three-point shots, which resulted in 54 points. They attempted $50 \%$ more two-point shots than three-point shots. How many three-point shots did they attempt?

(A) $10$

(B) $15$

(D) $25$

(E) $30$

AMC 10B, 2013, Problem 15

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$ ?

(A) $1$

(B) $\frac{6}{2}$

(D) $8$

(E) $10$

AMC 10A, 2012, Problem 1

Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
(A) $10$
(B) $15$
(C) $20$
(D) $25$
(E) $30$

AMC 10A, 2012, Problem 3

A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to 5 . How many units does the bug crawl altogether?
(A) $9$
(B) $11$
(C) $13$
(D) $14$
(E) $15$

AMC 10A, 2012, Problem 5

Last year $100$ adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was $4$ . What was the total number of cats and kittens received by the shelter last year?
(A) $150$
(B) $200$
(C) $250$
(D) $300$
(E) $400$

AMC 10A, 2012, Problem 6

The product of two positive numbers is $9$ . The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?
(A) $\frac{10}{3}$
(B) $\frac{20}{3}$
(C) $7$
(D) $\frac{15}{2}$
(E) $8$

AMC 10A, 2012, Problem 7

In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles
stays the same, what fraction of the marbles will be red?
(A) $\frac{2}{5}$
(B) $\frac{3}{7}$
(C) $\frac{4}{7}$
(D) $\frac{3}{5}$
(E) $\frac{4}{5}$

AMC 10A, 2012, Problem 8

The sums of three whole numbers taken in pairs are $12,17$, and $19 .$ What is the middle number?
(A) $4$
(B) $5$
(C) $6$
(D) $7$
$(\mathbf{E}) 8$

AMC 10A, 2012, Problem 10

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
(A) $5$
(B) $6$
(C) $8$
(D) $10$
(E) $12$

AMC 10A, 2012, Problem 13

An iterative average of the numbers $1,2,3,4$, and $5$ is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
(A) $\frac{31}{16}$
(B) $2$
(C) $\frac{17}{8}$
(D) $3$
(E) $\frac{65}{16}$

AMC 10A, 2012, Problem 14

Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
(A) $480$
(B) $481$
(C) $482$
(D) $483$
(E) $484$

AMC 10A, 2012, Problem 16

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
(A) $1,000$
(B) $1,250$
(C) $2,500$
(D) $5,000$
(E) $10,000$

AMC 10A, 2012, Problem 17

Let $a$ and $b$ be relatively prime positive integers with $a>b>0$ and $\frac{a^{3}-b^{3}}{(a-b)^{3}}=\frac{73}{3}$. What is $a-b$ ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

AMC 10A, 2012, Problem 19

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8: 00 \mathrm{AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50 \%$ of a house, quitting at $4: 00$ PM. On Tuesday, when Paula wasn't there, the two helpers painted only $24 \%$ of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
(A) 30
(B) 36
(C) 42
(D) 48
(E) 60

AMC 10B, 2012, Problem 1

Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all of the third-grade classrooms?
(A) $48$
(B) $56$
(C) $64$
(D) $72$
(E) $80$

AMC 10B, 2012, Problem 4

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2012, Problem 5

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10 \%$. She leaves a $15 \%$ tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
(A) $18$
(B) $20$
(C) $21$
(D) $22$
(E) $24$

AMC 10B, 2012, Problem 7

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
(A) $30$
(B) $36$
(C) $42$
(D) $48$
(E) $54$

AMC 10B, 2012, Problem 9

Two integers have a sum of 26 . When two more integers are added to the first two integers the sum is 41 . Finally when two more integers are added to the sum of the previous four integers the sum is $57 .$ What is the minimum number of odd integers among the 6 integers?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2012, Problem 10

How many ordered pairs of positive integers $(M, N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}$ ?
(A) $6$
(B) $7$
(C) $8$
(D) $9$
(E) $10$

AMC 10B, 2012, Problem 13

It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
(A) $36$
(B) $40$
(C) $42$
(D) $48$
(E) $52$

AMC 10B, 2012, Problem 17

Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
(A) $\frac{1}{8}$

(B) $\frac{1}{4}$

(C)$\frac{\sqrt{10}}{10}$

(D) $\frac{\sqrt{5}}{6}$

(E) $\frac{\sqrt{5}}{5}$

AMC 10A, 2011, Problem 1

A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent 100 text messages and talked for $30.5$ hours. How much did she have to pay?
(A) $24.00$
(B) $24.50$
(C) $25.50$
(D) $28.00$
(E) $30.00$

AMC 10A, 2011, Problem 2

A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
(A) $11$
(B) $12$
(C) $13$
(D) $14$
(E) $15$

AMC 10A, 2011, Problem 3

Suppose $[a b]$ denotes the average of $a$ and $b$, and ${a b c}$ denotes the average of $a, b$, and $c$. What is ${{1 1 0} [0 1 ] 0 }$ ?

(A) $\frac{2}{9}$
(B) $\frac{5}{18}$
(C) $\frac{1}{3}$
(D) $\frac{7}{18}$
(E) $\frac{2}{3}$

AMC 10A, 2011, Problem 4

Let $X$ and $Y$ be the following sums of arithmetic sequences:
$$

X=10+12+14+\cdots+100 \
Y=12+14+16+\cdots+102

$$
What is the value of $Y-X ?$
(A) $92$
(B) $98$
(C) $100$
(D) $102$
(E) $112$

AMC 10A, 2011, Problem 6

Set $A$ has 20 elements, and set $B$ has 15 elements. What is the smallest possible number of elements in $A \cup B$ ?
(A) $5$
(B) $15$
(C) $20$
(D) $35$
(E) $300$

AMC 10A, 2011, Problem 7

Which of the following equations does NOT have a solution?
(A) $(x+7)^{2}=0$
(B) $|-3 x|+5=0$
(C) $\sqrt{-x}-2=0$
(D) $\sqrt{x}-8=0$
$(\mathrm{E})|-3 x|-4=0$

AMC 10A, 2011, Problem 8

Last summer $30 \%$ of the birds living on Town Lake were geese, $25 \%$ were swans, $10 \%$ were herons, and $35 \%$ were ducks. What percent of the birds that were not swans were geese?
(A) $20$
(B) $30$
(C) $40$
(D) $50$
(E) $60$

AMC 10A, 2011, Problem 10

A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1 . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$ 17.71$. What was the cost of a pencil in cents?
(A) $7$
(B) $11$
(C) $17$
(D) $23$
(E) $77$

AMC 10A, 2011, Problem 11

Square $E F G H$ has one vertex on each side of square $A B C D$. Point $E$ is on $A B$ with $A E=7 \cdot E B$. What is the ratio of the area of $E F G H$ to the area of $A B C D ?$
(A) $\frac{49}{64}$
(B) $\frac{25}{32}$
(C) $\frac{7}{8}$
(D) $\frac{5 \sqrt{2}}{8}$
(E) $\frac{\sqrt{14}}{4}$

AMC 10A, 2011, Problem 12

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
(A) $13$
(B) $14$
(C) $15$
(D) $16$
(E) $17$

AMC 10A, 2011, Problem 15

Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.02$ gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
(A) $140$
(B) $240$
(C) $440$
(D) $640$
(E) $840$

AMC 10A, 2011, Problem 16

Which of the following is equal to $\sqrt{9-6 \sqrt{2}}+\sqrt{9+6 \sqrt{2}}$ ?
(A) $3 \sqrt{2}$
(B) $2 \sqrt{6}$
(C) $\frac{7 \sqrt{2}}{2}$
(D) $3 \sqrt{3}$
(E) $6$

AMC 10B, 2011, Problem 1

What is

2 + 4 + 6 1 + 3 + 5 - 1 + 3 + 5 2 + 4 + 6 ?

$\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} ?$

(A) $-1$
(B) $\frac{5}{36}$
(C) $\frac{7}{12}$
(D) $\frac{147}{60}$
(E) $\frac{43}{3}$

AMC 10B, 2011, Problem 2

Josanna's test scores to date are $90,80,70,60$, and 85 . Her goal is to raise here test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
(A) $80$
(B) $82$
(C) $85$
(D) $90$
(E) $95$

AMC 10B, 2011, Problem 4

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A<B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
(A) $\frac{A+B}{2}$
(B) $\frac{A-B}{2}$
(C) $\frac{B-A}{2}$
(D) $B-A$
$(\mathbf{E}) A+B$

AMC 10B, 2011, Problem 5

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was 161 . What is the correct value of the product of $a$ and $b$ ?
(A) $116$
(B) $161$
(C) $204$
(D) $214$
(E) $224$

AMC 10B, 2011, Problem 6

On Halloween Casper ate $\frac{1}{2}$ of his candies and then gave $2$ candies to his brother. The next day he ate $\frac{1}{2}$ of his remaining candies and then gave
$4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning?
(A) $30$
(B) $39$
(C) $48$
(D) $57$
(E) $66$

AMC 10B, 2011, Problem 12

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles The track has a width of 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
(A) $\frac{\pi}{3}$
(B) $\frac{2 \pi}{3}$
(C) $\pi$
(D) $\frac{4 \pi}{3}$
(E) $\frac{5 \pi}{3}$

AMC 10B, 2011, Problem 14

A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot?
(A) $52$
(B) $58$
(C) $62$
(D) $68$
(E) $70$

AMC 10A, 2015, Problem 19

What is the product of all the roots of the equation

5 | x | + 8 - - - - - - \sqrt = x 2 - 16 - - - - - - \sqrt

$\sqrt{5|x|+8}=\sqrt{x^{2}-16}$

(A) $-64$
(B) $-24$
(C) $-9$
(D) $24$
(E) $576$

AMC 10A, 2010, Problem 1

Mary's top book shelf holds five books with the following widths, in centimeters: $6, \frac{1}{2}, 1,2.5$, and $10$ . What is the average book width, in centimeters?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10A, 2010, Problem 3

Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
(A) $3$
(B) $13$
(C) $18$
(D) $25$
(E) $29$

AMC 10A, 2010, Problem 4

A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
(A) $50.2$
(B) $51.5$
(C) $52.4$
(D) $53.8$
(E) $55.2$

AMC 10A, 2010, Problem 8

Tony works $2$ hours a day and is paid $\$ 0.50$ per hour for each full year of his age. During a six month period Tony worked 50 days and earned $\$$
$630$. How old was Tony at the end of the six month period?
(A) $9$
(B) $11$
(C) $12$
(D) $13$
(E) $14$

AMC 10A, 2010, Problem 9

A palindrome, such as 83438 , is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit anc four-digit palindromes, respectively. What is the sum of the digits of $x$ ?
(A) $20$
(B) $21$
(C) $22$
(D) $23$
(E) $24$

AMC 10A, 2010, Problem 10

Marvin had a birthday on Tuesday, May 27 in the leap year 2008 . In what year will his birthday next fall on a Saturday?
(A) $2011$
(B) $2012$
(C) $2013$
(D) $2015$
(E) $2017$

AMC 10A, 2010, Problem 11

The length of the interval of solutions of the inequality $a \leq 2 x+3 \leq b$ is 10 . What is $b-a$ ?
(A) $6$
(B) $10$
(C) $15$
(D) $20$
(E) $30$

AMC 10A, 2010, Problem 12

Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds $0.1$ liters. How tall, in meters, should Logan make his tower?
(A) $0.04$
(B) $\frac{0.4}{\pi}$
(C) $0.4$
(D) $\frac{4}{\pi}$
(E) $4$

AMC 10A, 2010, Problem 13

Angelina drove at an average rate of $80 \mathrm{kmh}$ and then stopped 20 minutes for gas. After the stop, she drove at an average rate of $100 \mathrm{kmh}$. Altogether she drove $250 \mathrm{~km}$ in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
(A) $80 t+100\left(\frac{8}{3}-t\right)=250$
(B) $80 t=250$
(C) $100 t=250$
(D) $90 t=250$
(E) $80\left(\frac{8}{3}-t\right)+100 t=250$

AMC 10A, 2010, Problem 21

The polynomial $x^{3}-a x^{2}+b x-2010$ has three positive integer roots. What is the smallest possible value of $a$ ?
(A) $78$
(B) $88$
(C) $98$
(D) $108$
(E) $118$

AMC 10B, 2010, Problem 1

What is $100(100-3)-(100 \cdot 100-3)$ ?
(A) $-20,000$
(B) $-10,000$
(C) $-297$
(D) $-6$
(E) $0$

AMC 10B, 2010, Problem 2

Makarla attended two meetings during her 9 -hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
(A) $15$
(B) $20$
(C) $25$
(D) $30$
(E) $35$

AMC 10B, 2010, Problem 5

A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$

AMC 10B, 2010, Problem 7

Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
(A) 18
(B) 21
(C) 24
(D) 27
(E) 30

AMC 10B, 2010, Problem 8

A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of $9_{\text {th }}$ graders buys tickets costing a total of $\$ 48$, and a group of 10 th graders buys tickets costing a total of $\$ 64$. How many values for $x$ are possible?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2010, Problem 10

Lucky Larry's teacher asked him to substitute numbers for $a, b, c, d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a, b, c$, and $d$ were $1,2,3$, and 4 , respectively. What number did Larry substitute for $e$ ?
(A) $-5$
(B) $-3$
(C) $0$
(D) $3$
(E) $5$

AMC 10B, 2010, Problem 11

A shopper plans to purchase an item that has a listed price greater than $\$ 100$ and can use any one of the three coupons. Coupon A gives $15 \%$ off the listed price, Coupon B gives $\$ 30$ off the listed price, and Coupon C gives $25 \%$ off the amount by which the listed price exceeds $\$ 100$. Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y-x$ ?
(A) $50$
(B) $60$
(C) $75$
(D) $80$
(E) $100$

AMC 10B, 2010, Problem 12

At the beginning of the school year, $50 \%$ of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and $50 \%$ answered "No." At the end of the school year, $70 \%$ answered "Yes" and $30 \%$ answered "No." Altogether, $x \%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$ ?
(A) $0$
(B) $20$
(C) $40$
(D) $60$
(E) $80$

AMC 10B, 2010, Problem 13

What is the sum of all the solutions of $x=|2 x-| 60-2 x||$ ?
(A) $32$
(B) $60$
(C) $92$
(D) $120$
(E) $124$

AMC 10B, 2010, Problem 14

The average of the numbers $1,2,3, \cdots, 98,99$, and $x$ is $100 x$. What is $x$ ?
(A) $\frac{49}{101}$
(B) $\frac{50}{101}$
(C) $\frac{1}{2}$
(D) $\frac{51}{101}$
(E) $\frac{50}{99}$

AMC 10A, 2010, Problem 15

On a 50 -question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and $-1$ point for an incorrect answer. Jesse's total score on the contest was 99 . What is the maximum number of questions that Jesse could have answered correctly?
(A) $25$
(B) $27$
(C) $29$
(D) $31$
(E) $33$

AMC 10A, 2009, Problem 1

One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
(A) $7$
(B) $8$
(C) $9$
(D) $10$
(E) $11$

AMC 10A, 2009, Problem 4

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
(A) $15$
(B) $25$
(C) $35$
(D) $45$
(E) $55$

AMC 10A, 2009, Problem 3

Which of the following is equal to $1+\frac{1}{1+\frac{1}{1+1}}$ ?
(A) $\frac{5}{4}$
(B) $\frac{3}{2}$
(C) $\frac{5}{3}$
(D) $2$
(E) $3$

AMC 10A, 2009, Problem 4

Eric plans to compete in a triathlon. He can average 2 miles per hour in the $\frac{1}{4}$ mile swim and 6 miles per hour in the 3 -mile run. His goal is to finish the triathlon in 2 hours. To accomplish his goal what must his average speed in miles per hour, be for the 15 -mile bicycle ride?
(A) $\frac{120}{11}$
(B) $11$
(C) $\frac{56}{5}$
(D) $\frac{45}{4}$
(E) $12$

AMC 10A, 2009, Problem 5

What is the sum of the digits of the square of 111111111 ?
(A) $18$
(B) $27$
(C) $45$
(D) $63$
(E) $81$

AMC 10A, 2009, Problem 7

A carton contains milk that is $2 \%$ fat, an amount that is $40 \%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
(A) $\frac{12}{5}$
(B) $3$
(C) $\frac{10}{3}$
(D) $38$
(E) $42$

AMC 10A, 2009, Problem 8

Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50 \%$ discount as children. The two members of the oldest generation receive a $25 \%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $\$ 6.00$, is paying for everyone. How many dollars must he pay?
(A) $34$
(B) $36$
(C) $42$
(D) $46$
(E) $48$

AMC 10A, 2009, Problem 9

Positive integers $a, b$, and 2009, with $a<b<2009$, form a geometric sequence with an integer ratio. What is $a$ ?
(A) $7$
(B) $41$
(C) $49$
(D) $289$
(E) $2009$

AMC 10A, 2009, Problem 16

Let $a, b, c$, and $d$ be real numbers with $|a-b|=2,|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$ ?
(A) $9$
(B) $12$
(C) $15$
(D) $18$
(E) $24$

AMC 10A, 2009, Problem 18

At Jefferson Summer Camp, $60 \%$ of the children play soccer, $30 \%$ of the children swim, and $40 \%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
(A) $30 \%$
(B) $40 \%$
(C) $49 \%$
(D) $51 \%$
(E) $70 \%$

AMC 10B, 2009, Problem 1

Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
(A) $1$
(B) $2$
(C) $3$
(D) $4$
(E) $5$

AMC 10B, 2009, Problem 3

Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?
(A) $10$
(B) $12$
(C) $15$
(D) $18$
(E) $25$

AMC 10B, 2009, Problem 5

Twenty percent less than $60$ is one-third more than what number?
(A) $16$
(B) $30$
(C) $32$
(D) $36$
(E) $48$

AMC 10B, 2009, Problem 6

Kiana has two older twin brothers. The product of their three ages is $128$ . What is the sum of their three ages?
(A) $10$
(B) $12$
(C) $16$
(D) $18$
(E) $24$

AMC 10A, 2009, Problem 8

In a certain year the price of gasoline rose by $20 \%$ during January, fell by $20 \%$ during February, rose by $25 \%$ during March, and fell by $x \%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$
(A) $12$
(B) $17$
(C) $20$
(D) $25$
(E) $35$

AMC 10B, 2009, Problem 15

When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water?
(A) $\frac{2}{3} a+\frac{1}{3} b$
(B) $\frac{3}{2} a-\frac{1}{2} b$
(C) $\frac{3}{2} a+b$
(D) $\frac{3}{2} a+2 b$
(E) $3 a-2 b$

AMC 10B, 2009, Problem 19

A particular 12 -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1, it mistakenly displays a 9. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
(A) $\frac{1}{2}$
(B) $\frac{5}{8}$
(C) $\frac{3}{4}$
(D) $\frac{5}{6}$
(E) $\frac{9}{10}$

AMC 10A, 2008, Problem 1

A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
(A) $1: 50 \mathrm{PM}$
(B) 3:00 PM
(C) $3: 30 \mathrm{PM}$
(D) 4:30 PM
(E) 5:50 PM

AMC 10A, 2008, Problem 2

A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2: 1$. The ratio of the rectangle's length to its width is $2: 1$. What percent of the rectangle's area is inside the square?
(A) $12.5$
(B) $25$
(C) $50$
(D) $75$
(E) $87.5$

AMC 10A, 2008, Problem 3

For the positive integer $n$, let $\langle n\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4\rangle=1+2=3$ and $\langle 12\rangle=1+2+3+4+6=16$. What is $\langle\langle\langle 6\rangle\rangle\rangle$ ?
(A) $6$
(B) $12$
(C) $24$
(D) $32$
(E) $36$

AMC 10A, 2008, Problem 4

Suppose that $\frac{2}{3}$ of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as $\frac{1}{2}$ of 5 bananas?
(A) $2$
(B) $\frac{5}{2}$
(C) $3$
(D) $\frac{7}{2}$
(E) $4$

AMC 10A, 2008, Problem 4

Which of the following is equal to the product

8 4 \cdot 12 8 \cdot 16 12 \dots \dots 4 n + 4 4 n \dots \dots 2008 2004 ?

$\frac{8}{4} \cdot \frac{12}{8} \cdot \frac{16}{12} \cdots \cdots \frac{4 n+4}{4 n} \cdots \cdots \frac{2008}{2004} ?$

(A) $251$
(B) $502$
(C) $1004$
(D) $2008$
(E) $4016$

AMC 10A, 2008, Problem 6

A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
(A) $3$
(B) $4$
(C) $5$
(D) $6$
(E) $7$

AMC 10A, 2008, Problem 7

The fraction

( 3 2008 ) 2 - ( 3 2006 ) 2 ( 3 2007 ) 2 - ( 3 2005 ) 2

$\frac{\left(3^{2008}\right)^{2}-\left(3^{2006}\right)^{2}}{\left(3^{2007}\right)^{2}-\left(3^{2005}\right)^{2}}$

simplifies to which of the following?
(A) $1$
(B) $\frac{9}{4}$
(C) $3$
(D) $\frac{9}{2}$
(E) $9$

AMC 10A, 2008, Problem 8

Heather compares the price of a new computer at two different stores. Store $A$ offers $15 \%$ off the sticker price followed by a $\$ 90$ rebate, and store $B$ offers $25 \%$ off the same sticker price with no rebate. Heather saves $\$ 15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars?
(A) $750$
(B) $900$
(C) $1000$
(D) $1050$
(E) $1500$

AMC 10A, 2008, Problem 9

Suppose that

2 x 3 - x 6

$\frac{2 x}{3}-\frac{x}{6}$

is an integer. Which of the following statements must be true about $x$ ?
(A) It is negative.
(B) It is even, but not necessarily a multiple of 3 .
(C) It is a multiple of 3 , but not necessarily even.
(D) It is a multiple of 6, but not necessarily a multiple of 12 .
(E) It is a multiple of 12 .

AMC 10A, 2008, Problem 12

In a collection of red, blue, and green marbles, there are $25 \%$ more red marbles than blue marbles, and there are $60 \%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection?
(A) $2.85 r$
(B) $3 r$
(C) $3.4 r$
(D) $3.85 r$
(E) $4.25 r$

AMC 10A, 2008, Problem 13

Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$ ?
(A) $\left(\frac{1}{5}+\frac{1}{7}\right)(t+1)=1$
(B) $\left(\frac{1}{5}+\frac{1}{7}\right) t+1=1$
(C) $\left(\frac{1}{5}+\frac{1}{7}\right) t=1$
(D) $\left(\frac{1}{5}+\frac{1}{7}\right)(t-1)=1$
(E) $(5+7) t=1$

AMC 10A, 2008, Problem 15

Yesterday Han drove 1 hour longer than lan at an average speed 5 miles per hour faster than lan. Jan drove 2 hours longer than lan at an average speed 10 miles per hour faster than lan. Han drove 70 miles more than lan. How many more miles did Jan drive than lan?
(A) $120$
(B) $130$
(C) $140$
(D) $150$
(E) $160$

AMC 10B, 2008, Problem 3

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x \sqrt{x}}$ ?
(A) $x^{1 / 6}$
(B) $x^{1 / 4}$
(C) $x^{3 / 8}$
(D) $x^{1 / 2}$
(E) $x$

AMC 10B, 2008, Problem 5

For real numbers $a$ and $b$, define $a * b=(a-b)^{2}$. What is $(x-y)^{2} *(y-x)^{2}$ ?
(A) $0$
(B) $x^{2}+y^{2}$
(C) $2 x^{2}$
(D) $2 y^{2}$
(E) $4 x y$

AMC 10B, 2008, Problem 9

A quadratic equation $a x^{2}-2 a x+b=0$ has two real solutions. What is the average of these two solutions?
(A) $1$
(B) $2$
(C) $\frac{b}{a}$
(D) $\frac{2 b}{a}$
(E) $\sqrt{2 b-a}$

AMC 10B, 2008, Problem 18

Bricklayer Brenda would take nine hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together, they talk a lot, and their combined output decreases by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney?
(A) $500$
(B) $900$
(C) $950$
(D) $1000$
(E) $1900$

AMC 10A, 2007, Problem 1

One ticket to a show costs $\$ 20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25 \%$ discount. Pam buys 5 tickets using coupon that gives her a $30 \%$ discount. How many more dollars does Pam pay than Susan?
(A) $2$
(B) $5$
(C) $10$
(D) $15$
(E) $20$

AMC 10A, 2007, Problem 4

The larger of two consecutive odd integers is three times the smaller. What is their sum?
(A) $4$
(B) $8$
(C) $12$
(D) $16$
(E) $20$

AMC 10A, 2009, Problem 5

The school store sells $7$ pencils and $8$ notebooks for $\$ 4.15$. It also sells $5$ pencils and $3$ notebooks for $\$ 1.77$. How much do 16 pencils and 10 notebooks cost?
$(A) \$ 1.76$
(B) $\$ 5.84$
(C) $\$ 6.00$
(D) $\$ 6.16$
(E) $\$ 6.32$

AMC 10A, 2008, Problem 7

Last year Mr. Jon Q. Public received an inheritance. He paid $20 \%$ in federal taxes on the inheritance, and paid $10 \%$ of what he had left in state taxes. He paid a total of $\$ 10500$ for both taxes. How many dollars was his inheritance?
(A) $30000$
(B) $32500$
(C) $35000$
(D) $37500$
(E) $40000$

AMC 10A, 2007, Problem 9

Real numbers $a$ and $b$ satisfy the equations $3^{a}=81^{b+2}$ and $125^{b}=5^{a-3}$. What is $a b$ ?
(A) $-60$
(B) $-17$
(C) $9$
(D) $12$
(E) $60$

AMC 10A, 2007, Problem 10

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20 , the father is 48 years old, and the average age of the mother and children is 16 . How many children are in the family?
(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$

AMC 10A, 2007, Problem 13

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
(A) $\frac{2}{3}$
(B) $\frac{3}{4}$
(C) $\frac{4}{5}$
(D) $\frac{5}{6}$
(E) $\frac{7}{8}$

AMC 10A, 2007, Problem 20

Suppose that the number $a$ satisfies the equation $4=a+a^{-1}$. What is the value of $a^{4}+a^{-4}$ ?
(A) $164$
(B) $172$
(C) $192$
(D) $194$
(E) $212$

AMC 10B, 2007, Problem 2

Define the operation $\star$ by $a \star b=(a+b) b$. What is $(3 \star 5)-(5 \star 3) ?$
(A) $-16$
(B) $-8$
(C) $0$
(D) $8$
(E) $16$

AMC 10B, 2007, Problem 6

The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?
(A) $13$
(B) $14$
(C) $15$
(D) $16$
(E) $17$

AMC 10B, 2007, Problem 12

Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T / N ?$
(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$

AMC 10B, 2007, Problem 14

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40 \%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30 \%$ of the group are girls. How many girls were initially in the group?
(A) $4$
(B) $6$
(C) $8$
(D) $10$
(E) $12$

AMC 10B, 2007, Problem 16

A teacher gave a test to a class in which $10 \%$ of the students are juniors and $90 \%$ are seniors. The average score on the test was 84 . The juniors all received the same score, and the average score of the seniors was 83 . What score did each of the juniors receive on the test?
(A) $85$
(B) $88$
(C) $93$
(D) $94$
(E) $98$

AMC 10 Number Theory Questions - Year wise

American Mathematics contest 10 (AMC 10) - Number Theory problems

AMC 10A, 2021, Problem 10

Which of the following is equivalent to
$$
(2+3)\left(2^{2}+3^{2}\right)\left(2^{4}+3^{4}\right)\left(2^{8}+3^{8}\right)\left(2^{16}+3^{16}\right)\left(2^{32}+3^{32}\right)\left(2^{64}+3^{64}\right) ?
$$
(A) $3^{127}+2^{127}$
(B) $3^{127}+2^{127}+2 \cdot 3^{63}+3 \cdot 2^{63}$
(C) $3^{128}-2^{128}$
(D) $3^{128}+2^{128}$
(E) $5^{127}$

AMC 10A, 2021, Problem 11

For which of the following integers $b$ is the base- $b$ number $2021_{b}-221_{b}$ not divisible by $3$ ?
(A) $3$
(B) $4$
(C) $6$
(D) $7$
(E) $8$

AMC 10A, 2021, Problem 16

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
$$
1,2,2,3,3,3,4,4,4,4, \ldots, 200,200, \ldots, 200
$$
What is the median of the numbers in this list?
(A) $100.5$
(B) $134$
(C) $142$
(D) $150.5$
(E) $167$

AMC 10A, 2021, Problem 19

The area of the region bounded by the graph of
$$
x^{2}+y^{2}=3|x-y|+3|x+y|
$$
is $m+n \pi$, where $m$ and $n$ are integers. What is $m+n$ ?
(A) $18$
(B) $27$
(C) $36$
(D) $45$
(E) $54$

AMC 10B, 2021, Problem 1

How many integer values of $x$ satisfy $|x|<3 \pi ?$
(A) $9$
(B) $10$
(C) $18$
(D) $19$
(E) $20$

AMC 10B, 2021, Problem 13

Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32 d$ in base $n$ equals 263 , and the value of the numeral $324$ in base $n$ equals the value of the numeral $11 d 1$ in base six. What is $n+d$ ?
(A) $10$
(B) $11$
(C) $13$
(D) $15$
(E) $16$

AMC 10B, 2021, Problem 16

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357,89$ , and $5$ are all uphill integers, but $32,1240$, and $466$ are not. How many uphill integers are divisible by $15$ ?
(A) $4$
(B) $5$
(C) $6$
(D) $7$
(E) $8$

AMC 10B, 2021, Problem 19

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer in $S$ is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set $S$ is 72 greater than the least integer in $S$. What is the average value of all the integers in the set $S ?$
(A) $36.2$
(B) $36.4$
(C) $36.6$
(D) $36.8$
(E) $37$

AMC 10A, 2020, Problem 4

A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $0.50$ per mile, and her only expense is gasoline at $2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

(A) $20$

(B) $22$

(D)$25$

(E) $26$

AMC 10A, 2020, Problem 6

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

(A)$80$

(B)$100$

(C)$125$

(D)$200$

(E) $500$

AMC 10A, 2020, Problem 8

What is the value of

$1+2+3-4+5+6+7-8+\cdots+197+198+199-200?$

(A) $9,800$

(B)$9,900 $

(D) $10,100$

(E)$10,200$

AMC 10A, 2020, Problem 9

(A) $9$

(B)$18$

(C)$27$

(D)$36$

(E)$77$

AMC 10A, 2020, Problem 17

Define $P(x) =(x-1^2)(x-2^2)\cdots(x-100^2)$.How many integers $n$ are there such that $P(n)\leq 0$?

(A)$4900$

(B)$4950$

(C)$5000$

(D)$5050$

(E) $5100$

AMC 10A, 2020, Problem 21

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that $\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}$ .What is $k?$

(A) $117$

(B)$136$

(C)$137$

(D)$273 $

(E)$306$

AMC 10A, 2020, Problem 22

For how many positive integers $n \le 1000$ is $\left\lfloor \frac{998}{n} \right\rfloor+\left\lfloor \frac{999}{n} \right\rfloor+\left\lfloor \frac{1000}{n}\right \rfloor$ not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

(A) $22$

(B)$23$

(C)$24$

(D)$25$

(E)$26$

AMC 10A, 2020, Problem 24

Let $n$ be the least positive integer greater than $1000$ for which $\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60$.What is the sum of the digits of $n$?

(A) $12$

(B)$15$

(C)$18$

(D)$21$

(E)$24$

AMC 10B, 2020, Problem 24

How many positive integers $n$ satisfy
$$
\frac{n+1000}{70}=\lfloor\sqrt{n}\rfloor ?
$$
(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
(A) $2$
(B) $4$
(C) $6$
(D) $30$
(E) $32$

AMC 10B, 2020, Problem 25

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product $n=f_{1} \cdot f_{2} \ldots f_{k}$
where $k \geq 1$, the $f_{i}$ are integers strictly greater than 1 , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as $6,2 \cdot 3$, and $3 \cdot 2$, so $D(6)=3$. What is $D(96)$ ?
(A) $112$
(B) $128$
(C) $144$
(D) $172$
(E) $184$

AMC 10A, 2019, Problem 5

What is the greatest number of consecutive integers whose sum is $45$ ?
(A) $9$
(B) $25$
(C) $45$
(D) $90$
(E) $120$

AMC 10A, 2019, Problem 9

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers?
(A) $995$
(B) $996$
(C) $997$
(D) $998$
(E) $999$

AMC 10A, 2019, Problem 15

A sequence of numbers is defined recursively by $a_{1}=1, a_{2}=\frac{3}{7}$, and
$$
a_{n}=\frac{a_{n-2} \cdot a_{n-1}}{2 a_{n-2}-a_{n-1}}
$$
for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$ ?
(A) $2020$
(B) $40394
(C) 46057$
(D) $6061$
(E) $8078$

AMC 10A, 2019, Problem 18

For some positive integer $k$, the repeating base- $k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0 . \overline{23}_{k}=0.232323 \ldots k$. What is $k ?$
(A) $13$
(B) $14$
(C) $15$
(D) $16$
(E) $17$

AMC 10A, 2019, Problem 19

What is the least possible value of
$$
(x+1)(x+2)(x+3)(x+4)+2019
$$
where $x$ is a real number?
(A) $2017$
(B) $2018$
(C) $2019$
(D) $2020
(E) $2021$

AMC 10A, 2019, Problem 25

For how many integers $n$ between 1 and 50 , inclusive, is
$$
\frac{\left(n^{2}-1\right) !}{(n !)^{n}}
$$
an integer? (Recall that $0 !=1$.)
(A) $31$
(B) $32$
(C) $33$
(D) $34$
(E) $35$

AMC 10B, 2019, Problem 6

There is a positive integer $n$ such that $(n+1) !+(n+2) !=n ! \cdot 440$. What is the sum of the digits of $n$ ?
(A) $3$
(B) $8$
(C) $10$
(D) $11$
(E) $12$

AMC 10B, 2019, Problem 9

The function $f$ is defined by
$$
f(x)=\lfloor|x|\rfloor-|\lfloor x\rfloor|
$$
for all real numbers $x$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f ?$
(A) ${-1,0}$
(B) The set of nonpositive integers
(C) ${-1,0,1}$
(D) ${0}$
(E) The set of nonnegative integers

AMC 10B, 2019, Problem 12

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$ ?
(A) $11$
(B) $14$
(C) $22$
(D) $23$
(E) $27$

AMC 10B, 2019, Problem 19

Let $S$ be the set of all positive integer divisors of $100,000$ . How many numbers are the product of two distinct elements of $S ?$
(A) $98$
(B) $100$
(C) $117$
(D) $119$
(E) $121$

AMC 10B, 2019, Problem 24

Define a sequence recursively by $x_{0}=5$ and
$$
x_{n+1}=\frac{x_{n}^{2}+5 x_{n}+4}{x_{n}+6}
$$
for all nonnegative integers $n$. Let $m$ be the least positive integer such that
$$
x_{m} \leq 4+\frac{1}{2^{20}}
$$
In which of the following intervals does $m$ lie?
(A) $[9,26]$
(B) $[27,80]$
(C) $[81,242]$
(D) $[243,728]$
(E) $[729, \infty)$

AMC 10A, 2018, Problem 7

For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot\left(\frac{2}{5}\right)^{n}$ an integer?
(A) $3$
(B) $4$
(C) $6$
(D) $8$
(E) $9$

AMC 10A, 2018, Problem 14

What is the greatest integer less than or equal to
$$
\frac{3^{100}+2^{100}}{3^{96}+2^{96}} ?
$$
(A) $80$
(B) $81$
(C) $96$
(D) $97$
(E) $625$

AMC 10A, 2018, Problem 18

How many nonnegative integers can be written in the form
$$
a_{7} \cdot 3^{7}+a_{6} \cdot 3^{6}+a_{5} \cdot 3^{5}+a_{4} \cdot 3^{4}+a_{3} \cdot 3^{3}+a_{2} \cdot 3^{2}+a_{1} \cdot 3^{1}+a_{0} \cdot 3^{0},
$$
where $a_{i} \in{-1,0,1}$ for $0 \leq i \leq 7$ ?
(A) $512$
(B) $729$
(C) $1094$
(D) $3281$
(E) $59,048$

AMC 10A, 2018, Problem 22

Let $a, b, c$, and $d$ be positive integers such that $gcd(a, b)=24, gcd(b, c)=36, gcd(c, d)=54$, and $70<gcd(d, a)<100$. Which of the following must be a divisor of $a$ ?
(A) $5$
(B) $7$
(C) $11$
(D) $13$
(E) $17$

AMC 10A, 2018, Problem 25

For a positive integer $n$ and nonzero digits $a, b$, and $c$, let $A_{n}$ be the $n$ -digit integer each of whose digits is equal to $a$; let $B_{n}$ be the $n$ -digit integer each of whose digits is equal to $b$, and let $C_{n}$ be the $2 n$ -digit (not $n$ -digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a+b+c$ for which there are at least two values of $n$ such that $C_{n}-B_{n}=A_{n}^{2}$ ?
(A) $12$
(B) $144
(C) $16$
(D) $18$
(E) $20$

AMC 10B, 2018, Problem 5

How many subsets of ${2,3,4,5,6,7,8,9}$ contain at least one prime number?
(A) $128$
(B) $192$
(C) $224$
(D) $240$
(E) $256$

AMC 10B, 2018, Problem 11

Which of the following expressions is never a prime number when $p$ is a prime number?
(A) $p^{2}+16$
(B) $p^{2}+24$
(C) $p^{2}+26$
(D) $p^{2}+46$
(E) $p^{2}+96$

AMC 10B, 2018, Problem 13

How many of the first 2018 numbers in the sequence $101,1001,10001,100001, \ldots$ are divisible by $101$ ?
(A) $253$
(B) $504$
(C) $505$
(D) $506$
(E) $1009$

AMC 10B, 2018, Problem 14

A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
(A) $202$
(B) $223$
(C) $224$
(D) $225$
(E) $234$

AMC 10B, 2018, Problem 16

Let $a_{1}, a_{2}, \ldots, a_{2018}$ be a strictly increasing sequence of positive integers such that
$$
a_{1}+a_{2}+\cdots+a_{2018}=2018^{2018}
$$
What is the remainder when $a_{1}^{3}+a_{2}^{3}+\cdots+a_{2018}^{3}$ is divided by $6$ ?
(A) $0$
(B) $1$
(C) $2$
(D) $3$
(E) $4$

AMC 10B, 2018, Problem 20

A function $f$ is defined recursively by $f(1)=f(2)=1$ and
$$
f(n)=f(n-1)-f(n-2)+n
$$
for all integers $n \geq 3$. What is $f(2018)$ ?
(A) $2016$
(B) $2017$
(C) $2018$
(D) $2019$
(E) $2020$

AMC 10B, 2018, Problem 21

Mary chose an even 4 -digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote 323 as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$ ?
(A) $324$
(B) $330$
(C) $340$
(D) $361$
(E) $646$

AMC 10B, 2018, Problem 23

How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$
a \cdot b+63=20 \cdot lcm(a, b)+12 \cdot gcd(a, b)
$$
where $gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $lcm(a, b)$ denotes their least common multiple?
(A) $0$
(B) $2$
(C) $4$
(D) $6$
(E) $8$

AMC 10B, 2018, Problem 25

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^{2}+10,000\lfloor x\rfloor=10,000 x$ ?
(A) $197$
(B) $198$
(C) $199$
(D) $200$
(E) $201$

AMC 10B, 2018, Problem 12

Let $S$ be a set of points $(x, y)$ in the coordinate plane such that two of the three quantities $3, x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S$ ?
(A) a single point
(B) two intersecting lines
(C) three lines whose pairwise intersections are three distinct points
(D) a triangle
(E) three rays with a common endpoint

AMC 10A, 2017, Problem 13

Define a sequence recursively by $F_{0}=0, F_{1}=1$, and $F_{n}=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by 3, for all $n \geq 2$. Thus the sequence starts $0,1,1,2,0,2, \ldots$ What is $F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}$ ?
(A) $6$
(B) $7$
(C) $8$
(D) $9$
(E) $10$

AMC 10A, 2017, Problem 17

Distinct points $P, Q, R, S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $P Q$ and $R S$ are irrational numbers. What is the greatest possible value of the ratio $\frac{P Q}{R S} ?$
(A) $3$
(B) $5$
(C) $3 \sqrt{5}$
(D) $7$
(E) $5 \sqrt{2}$

AMC 10A, 2017, Problem 20

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507)=13$. For a particular positive integer $n, S(n)=1274$. Which of the following could be the value of $S(n+1)$ ?
(A) $1$
(B) $3$
(C) $12$
(D) $1239$
(E) $1265$

AMC 10A, 2017, Problem 23

How many triangles with positive area have all their vertices at points $(i, j)$ in the coordinate plane, where $i$ and $j$ are integers between 1 and 5, inclusive?
(A) $2128$
(B) $2148$
(C) $2160$
(D) $2200$
(E) $2300$

AMC 10A, 2017, Problem 25

How many integers between 100 and 999 , inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 211 have this property.
(A) $226$
(B) $243$
(C) $270$
(D) $469$
(E) $486$

AMC 10B, 2017, Problem 14

An integer $N$ is selected at random in the range $1 \leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by 5 is 1 ?
(A) $\frac{1}{5}$
(B) $\frac{2}{5}$
(C) $\frac{3}{5}$
(D) $\frac{4}{5}$
(E) $1$

AMC 10B, 2017, Problem 16

How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0 ?
(A) $469$
(B) $471$
(C) $475$
(D) $478$
(E) $481$

AMC 10B, 2017, Problem 23

Let $N=123456789101112 \ldots 4344$ be the 79 -digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when $N$ is divided by 45 ?
(A) $1$
(B) $4$
(C) $9$
(D) $18$
(E) $44$

AMC 10B, 2017, Problem 25

Last year Isabella took $7$ math tests and received 7 different scores, each an integer between $91$ and 100 , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$ . What was her score on the sixth test?
(A) $92$
(B) $94$
(C) $96$
(D) $98$
(E) $100$

AMC 10A, 2016, Problem 4

The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by

$rem(x, y)=x-y \mid \frac{x}{y}\rfloor$

where $\left[\frac{x}{y}]\right.$ denotes the greatest integer less than or equal to $\frac{x}{y}$. What is the value of $rem\left(\frac{3}{8},-\frac{2}{5}\right) ?$
(A) $-\frac{3}{8}$
(B) $-\frac{1}{40}$
(C) $0$
(D) $\frac{3}{8}$
(E) $\frac{31}{40}$

AMC 10A, 2016, Problem 9

A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$ ?
(A) $6$
(B) $7$
(C) $8$
(D) $9$
(E) $10$

AMC 10A, 2016, Problem 17

Let $N$ be a positive multiple of 5 . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N)<\frac{321}{400}$ ?
(A) $12$
(B) $14$
(C) $16$
(D) $18$
(E) $20$

AMC 10A, 2016, Problem 20

For some particular value of $N$, when $(a+b+c+d+1)^{N}$ is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables $a, b, c$, and $d$, each to some positive power. What is $N$ ?
(A) $9$
(B) $14$
(C) $16$
(D) $17$
(E) $19$

AMC 10A, 2016, Problem 22

For some positive integer $n$, the number $110 \mathrm{n}^{3}$ has 110 positive integer divisors, including 1 and the number $110 \mathrm{n}^{3}$. How many positive integer divisors does the number $81 \mathrm{n}^{4}$ have?
(A) $110$
(B) $191$
(C) $261$
(D) $325$
(E) $425$

AMC 10A, 2016, Problem 25

How many ordered triples $(x, y, z)$ of positive integers satisfy $lcm(x, y)=72, lcm(x, z)=600$ and $lcm(y, z)=900 ?$
(A) $15$
(B) $16$
(C) $24$
(D) $27$
(E) $64$

AMC 10B, 2016, Problem 6

AMC 10B, 2016, Problem 13

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?
(A) $25$
(B) $40$
(C) $64$
(D) $100$
(E) $160$

AMC 10B, 2016, Problem 24

How many four-digit integers $a b c d$, with $a \neq 0$, have the property that the three two-digit integers $a b<b c<c d$ form an increasing arithmetic sequence? One such number is 4692 , where $a=4, b=6, c=9$, and $d=2$.
(A) 9
(B) 15
(C) 16
(D) 17
(E) 20

AMC 10B, 2016, Problem 25

Let $f(x)=\sum_{k=2}^{10}(\lfloor k x\rfloor-k\lfloor x\rfloor)$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to $r$.

How many distinct values does $f(x)$ assume for $x \geq 0 ?$

(A) 32
(B) 36
(C) 45
(D) 46
(E) infinitely many

AMC 10A, 2015, Problem 18

Hexadecimal (base-16) numbers are written using numeric digits 0 through 9 as well as the letters $A$ through $F$ to represent 10 through $15 .$ Among the first 1000 positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$ ?
(A) 17
(B) 18
(C) 19
(D) 20
(E) 21

AMC 10A, 2015, Problem 23

The zeroes of the function $f(x)=x^{2}-a x+2 a$ are integers. What is the sum of the possible values of $a ?$
(A) 7
(B) 8
(C) 16
(D) 17
(E) 18

AMC 10A, 2015, Problem 25

Let $S$ be a square of side length 1 . Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b \pi}{c}$, where $a, b$, and $c$ are positive integers with $gcd(a, b, c)=1 .$ What is $a+b+c$ ?
(A) 59
(B) 60
(C) 61
(D) 62
(E) 63

AMC 10B, 2015, Problem 10

What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$ ?
(A) It is a negative number ending with a 1.
(B) It is a positive number ending with a 1 .
(C) It is a negative number ending with a 5 .
(D) It is a positive number ending with a $5 .$
(E) It is a negative number ending with a 0 .

AMC 10B, 2015, Problem 14

Let $a, b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$ ?
(A) $15$
(B) $15.5$
(C) $16$
(D) $16.5$
(E) $17$

AMC 10B, 2015, Problem 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$ ?
(A) $9$
(B) $11$
(C) $12$
(D) $13$
(E) $15$

AMC 10B, 2015, Problem 23

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n !$ ends in $k$ zeros and the decimal representation of $(2 n)$. ends in $3 k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$ ?
(A) $7$
(B) $8$
(C) $9$
(D) $10$
(E) $11$

AMC 10A, 2014, Problem 20

The product $(8)(888 \ldots 8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of 1000 . What is $k$ ?
(A) 901
(B) 911
(C) 919
(D) 991
(E) 999

AMC 10A, 2014, Problem 24

A sequence of natural numbers is constructed by listing the first 4 , then skipping one, listing the next 5 , skipping 2 , listing 6 , skipping 3 , and, on the $n$ th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the 500,000 th number in the sequence?
(A) 996,506
(B) 996,507
(C) 996,508
(D) 996,509
(E) 996,510

AMC 10A, 2014, Problem 25

The number $5^{867}$ is between $2^{2013}$ and $2^{2014} .$ How many pairs of integers $(m, n)$ are there such that $1 \leq m \leq 2012$ and
$$
5^{n}<2^{m}<2^{m+2}<5^{n+1} ?
$$
(A) 278
(B) 279
(C) 280
(D) 281
(E) 282

AMC 10B, 2014, Problem 12

The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?
(A) $125,875,000$
(B) $201,400,000$
(C) $251,750,000$
(D) $402,800,000$
(E) $503,500,000$

AMC 10B, 2014, Problem 14

Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abe miles was displayed on the odometer, where $a b c$ is a 3-digit number with $a \geq 1$ and $a+b+c \leq 7$. At the end of the trip, the odometer showed $c b a$ miles. What is $a^{2}+b^{2}+c^{2} ?$
(A) 26
(B) 27
(C) 36
(D) 37
(E) 41

AMC 10B, 2014, Problem 17

What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$ ?
(A) $2^{1002}$
(B) $2^{1003}$
(C) $2^{1004}$
(D) $2^{1005}$
(E) $2^{1006}$

AMC 10B, 2014, Problem 20

For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative?
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

AMC 10A, 2013, Problem 13

How many three-digit numbers are not divisible by 5 , have digits that sum to less than 20 , and have the first digit equal to the third digit?
(A) 52
(B) 60
(C) 66
(D) 68
(E) 70

AMC 10A, 2013, Problem 19

In base 10, the number 2013 ends in the digit 3 . In base 9 , on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit 6 . For how many positive integers $b$ does the base- $b$ -representation of 2013 end in the digit 3 ?
(A) 6
(B) 9
(C) 13
(D) 16
(E) 18

AMC 10B, 2013, Problem 4

When counting from 3 to 201,53 is the $51^{n t}$ number counted. When counting backwards from 201 to 3,53 is the $n^{t h}$ number counted. What is $n$ ?
(A) 146
(B) 147
(C) 148
(D) 149
(E) 150

AMC 10B, 2013, Problem 5

Positive integers $a$ and $b$ are each less than 6 . What is the smallest possible value for $2 \cdot a-a \cdot b$ ?
(A) $-20$
(B) $-15$
(C) $-10$
(D) 0
(E) 2

AMC 10B, 2013, Problem 9

Three positive integers are each greater than 1 , have a product of 27000 , and are pairwise relatively prime. What is their sum?
(A) 100
(B) 137
(C) 156
(D) 160
(E) 165

AMC 10B, 2013, Problem 18

The number 2013 has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than 2013 but greater than 1000 have this property?
(A) 33
(B) 34
(C) 45
(D) 46
(E) 58

AMC 10B, 2013, Problem 20

The number 2013 is expressed in the form
$$
2013=\frac{a_{1} ! a_{2} ! \ldots a_{m} !}{b_{1} ! b_{2} ! \ldots b_{n} !}
$$
where $a_{1} \geq a_{2} \geq \cdots \geq a_{m}$ and $b_{1} \geq b_{2} \geq \cdots \geq b_{n}$ are positive integers and $a_{1}+b_{1}$ is as small as possible. What is $\left|a_{1}-b_{1}\right| ?$
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

AMC 10B, 2013, Problem 14

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$ ?
(A) 55
(B) 89
(C) 104
(D) 144
(E) 273

AMC 10B, 2013, Problem 18

AMC 10B, 2013, Problem 24

A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including 1 and $m$ ) such that the sum of the four divisors is equal to $n$. How many numbers in the set ${2010,2011,2012, \ldots, 2019}$ are nice?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

AMC 10B, 2013, Problem 25

Bernardo chooses a three-digit positive integer $N$ and writes both its base- 5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245 , and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2 N$ ?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25

AMC 10A, 2012, Problem 24

Let $a, b$, and $c$ be positive integers with $a \geq b \geq c$ such that $a^{2}-b^{2}-c^{2}+a b=2011$ and $a^{2}+3 b^{2}+3 c^{2}-3 a b-2 a c-2 b c=-1997$.
What is $a ?$
(A) 249
(B) 250
(C) 251
(D) 252
(E) 253

AMC 10B, 2012, Problem 8

What is the sum of all integer solutions to $1<(x-2)^{2}<25$ ?
(A) 10
(B) 12
(C) 15
(D) 19
(E) 25

AMC 10B, 2012, Problem 10

How many ordered pairs of positive integers $(M, N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N} ?$
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

AMC 10B, 2012, Problem 20

Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000 . Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$ ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

AMC 10A, 2011, Problem 13

How many even integers are there between $200$ and $700$ whose digits are all different and come from the set ${1,2,5,7,8,9} ?$
(A) 12
(B) 20
(C) 72
(D) 120
(E) 200

AMC 10A, 2011, Problem 17

In the eight term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is 5 and the sum of any three consecutive terms is 30 . What is $A+H$ ?
(A) 17
(B) 18
(C) 25
(D) 26
(E) 43

AMC 10A, 2011, Problem 19

In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011 , with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
(A) 42
(B) 47
(C) 52
(D) 57
(E) 62

AMC 10A, 2011, Problem 23

Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1,3,4,6,7,9, \ldots$ $, 997,999,1000$
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
(A) 37
(B) 242
(C) 365
(D) 728
(E) 998

AMC 10A, 2011, Problem 25

Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called $n$ -ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
(A) 1500
(B) 1560
(C) 2320
(D) 2480
(E) 2500

AMC 10B, 2011, Problem 10

Consider the set of numbers ${1,10,10^{2}, 10^{3}, \ldots, 10^{10}}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
(A) 1
(B) 9
(C) 10
(D) 11
(E) 101

AMC 10B, 2011, Problem 21

Brian writes down four integers $w>x>y>z$ whose sum is 44 . The pairwise positive differences of these numbers are $1,3,4,5,6$, and 9 What is the sum of the possible values for $w$ ?
(A) 16
(B) 31
(C) 48
(D) 62
(E) 93

AMC 10B, 2011, Problem 23

What is the hundreds digit of $2011^{2011}$ ?
(A) 1
(B) 4
(C) 5
(D) 6
(E) 9

AMC 10B, 2011, Problem 24

A lattice point in an $x y$ -coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y=m x+2$ passes through no lattice point with $0<x \leq 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$ ?
(A) $\frac{51}{101}$
(B) $\frac{50}{99}$
(C) $\frac{51}{100}$
(D) $\frac{52}{101}$
(E) $\frac{13}{25}$

AMC 10A, 2010, Problem 9

A palindrome, such as 83438 , is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x ?$
(A) 20
(B) 21
(C) 22
(D) 23
(E) 24

AMC 10A, 2010, Problem 25

Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n=55$, then his sequence contains 5 numbers:
$$
\begin{aligned}
& 55 \
55-7^{2} &=6 \
6-2^{2} &=2 \
2-1^{2} &=1 \
1-1^{2} &=0
\end{aligned}
$$
Let $N$ be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of $N$ ?
(A) $1$
(B) $3$
(C) $5$
(D) $7$
(E) $9$

AMC 10A, 2009, Problem 5

What is the sum of the digits of the square of 1111111111 ?
(A) 18
(B) 27
(C) 45
(D) 63
(E) 81

AMC 10A, 2009, Problem 13

Suppose that $P=2^{m}$ and $Q=3^{n}$. Which of the following is equal to $12^{m n}$ for every pair of integers $(m, n) ?$
(A) $P^{2} Q$
(B) $P^{n} Q^{m}$
(C) $P^{n} Q^{2 m}$
(D) $P^{2 m} Q^{n}$
(E) $P^{2 n} Q^{m}$

AMC 10A, 2009, Problem 25

For $k>0$, let $I_{k}=10 \ldots 064$, where there are $k$ zeros between the 1 and the 6 . Let $N(k$ ) be the number of factors of 2 in the prime factorization of $I_{k}$. What is the maximum value of $N(k)$ ?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

AMC 10A, 2009, Problem 21

What is the remainder when $3^{0}+3^{1}+3^{2}+\cdots+3^{2009}$ is divided by 8 ?
(A) 0
(B) 1
(C) 2
(D) 4
(E) 6

AMC 10A, 2008, Problem 24

Let $k=2008^{2}+2^{2008} \cdot$ What is the units digit of $k^{2}+2^{k}$ ?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8

AMC 10B, 2008, Problem 13

For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the $2008^{\text {th }}$ term of the sequence?
(A) 2008
(B) 4015
(C) 4016
(D) $4,030,056$
(E) $4,032,064$

AMC 8 Geometry Questions - Year wise

American Mathematics contest 8 (AMC 8) - Geometry problems

Try these AMC 8 Geometry Questions and check your knowledge!

AMC 8, 2025, Problem 1

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4 \times 4$ grid is covered by the star?

(A) 40
(B) 50
(C) 60
(D) 75
(E) 80

AMC 8, 2025, Problem 8

Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of 18 square centimeters. What is the volume of the cube in cubic centimeters?

(A) $3 \sqrt{3}$
(B) 6
(C) 9
(D) $6 \sqrt{3}$
(E) $9 \sqrt{3}$

AMC 8, 2025, Problem 10

In the figure below, $A B C D$ is a rectangle with sides of length $A B=5$ inches and $A D=3$ inches. Rectangle $A B C D$ is rotated $90^{\circ}$ clockwise around the midpoint of side $D C$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?

(A) 21
(B) 22.25
(C) 23
(D) 23.75
(E) 25

AMC 8, 2025, Problem 12

The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?

(A) $3 \pi$
(B) $4 \pi$
(C) $5 \pi$
(D) $6 \pi$
(E) $8 \pi$

AMC 8, 2025, Problem 18

The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R$, in units, of the circle on the right?

(A) $\sqrt{2}$
(B) 2
(C) $2 \sqrt{2}$
(D) 4
(E) $4 \sqrt{2}$

AMC 8, 2025, Problem 24

In trapezoid $A B C D$, angles $B$ and $C$ measure $60^{\circ}$ and $A B=D C$. The side lengths are all positive integers, and the perimeter of $A B C D$ is 30 units. How many non-congruent trapezoids satisfy all of these conditions?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

AMC 8, 2023, Problem 2

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?

(A)

(B)

(C)

(D)

(E)

AMC 8, 2023, Problem 7

A rectangle, with sides parallel to the x-axis and y-axis, has opposite vertices located at (15,3) and (16,5). A line is drawn through points A(0,0) and B(3,1). Another line is drawn through points C(0,10) and D(2,9). How many points on the rectangle lie on at least one of the two lines?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

AMC 8, 2023, Problem 12

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

(A) $\frac{1}{4}$
(B) $\frac{11}{36}$
(C) $\frac{1}{3}$
(D) $\frac{19}{36}$
(E) $\frac{5}{9}$

AMC 8, 2023, Problem 17

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of Q?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

AMC 8, 2023, Problem 19

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $ \frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

(A) 1: 3
(B) 3: 8
(C) 5: 12
(D) 7: 16
(E) 4: 9

AMC 8, 2023, Problem 24

Isosceles triangle A B C has equal side lengths A B and B C. In the figures below, segments are drawn parallel to $\overline{A C}$ so that the shaded portions of /( \triangle A B C /) have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height h of $\triangle A B C ?$

(A) 14.6
(B) 14.8
(C) 15
(D) 15.2
(E) 15.4

AMC 8, 2022, Problem 1

The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

(A) 10
(B) 12
(C) 13
(D) 14
(E) 15

AMC 8, 2022, Problem 4

The letter $\mathbf{M}$ in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?

AMC 8, 2022, Problem 18

The midpoints of the four sides of a rectangle are $(-3,0),(2,0),(5,4)$, and $(0,4)$. What is the area of the rectangle?
(A) 20
(B) 25
(C) 40
(D) 50
(E) 80

AMC 8, 2022, Problem 24

The figure below shows a polygon $A B C D E F G H$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $A H=E F=8$ and $G H=14$. What is the volume of the prism?

(A) 112
(B) 128
(C) 192
(D) 240
(E) 288

AMC 8, 2020, Problem 18

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline {FE}$ as shown in the figure. Let $DA = 16$, and let $FD = AE = 9$.What is the area of $ABCD$?

(A) $240$ (B) $248$ (C) $256$ (D) $264$ (E) $272$.

AMC 8, 2019, Problem 2

Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is $5$ feet, what is the area in square feet of rectangle $ABCD$?

(A) $45$ (B) $75$ (C) $100$ (D) $125$ (E) $150$.

AMC 8, 2019, Problem 4

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?

(A) $60$ (B) $90$ (C) $105$ (D) $120$ (E) $144$.

AMC 8, 2019, Problem 24

In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?

(A) $24$ (B) $30$ (C) $32$ (D) $36$ (E) $40$.

AMC 8, 2018, Problem 4

The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?

(A) $12$ (B) $12.5$ (C) $13$ (D) $13.5$ (E) $14$.

AMC 8, 2018, Problem 9

Bob is tiling the floor of his $12$ foot by $16$ foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?

(A) $48$ (B) $87$ (C) $91$ (D) $96$ (E) $120$.

AMC 8, 2018, Problem 15

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

(A) $ \frac{1}{4}$ (B) $\frac{1}{3}$ (C) $\frac{1}{2}$ (D) $1$ (E) $\frac{\pi}{2}$

AMC 8, 2018, Problem 20

In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$

(A)$ \frac{4}{9}$ (B) $\frac{1}{2}$ (C)$ \frac{5}{9}$ (D) $ \frac{3}{5}$ (E)$ \frac{2}{3}$.

AMC 8, 2018, Problem 22

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$

(A) $ 100 $ (B) $108$ (C) $120$ (D) $135$ (E) $144$.

AMC 8, 2018, Problem 23

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

(A) $ \frac{2}{7}$ (B) $ \frac{5}{42}$ (C)$ \frac{11}{14} $ (D) $\frac{5}{7}$ (E) $\frac{6}{7}$.

AMC 8, 2018, Problem 24

In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$

(A) $\frac{5}{4}$ (B) $ \frac{4}{3} $ (C) $ \frac{3}{2} $ (D) $ \frac{25}{16}$ (E) $\frac{9}{4}$.

AMC 8, 2017, Problem 16

In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?

(A) $\frac{3}{4}$ (B) $\frac{3}{2}$ (C) $2$ (D) $\frac{12}{5}$ (E)$\frac{5}{2}$.

AMC 8, 2017, Problem 18

In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of the quadrilateral $ABCD$?

(A) $12$ (B) $24$ (C) $26$ (D) $30$ (E) $36$.

AMC 8, 2017, Problem 22

In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

(A) $\frac{7}{6}$ (B) $\frac{13}{5}$ (C) $\frac{59}{18}$ (D) $\frac{10}{3}$ (E) $\frac{60}{13}$.

AMC 8, 2017, Problem 25

In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length $2$, and $m\angle TUS = 60^{\circ}$. Arcs ${TR}$ and ${SR}$ are each one-sixth of a circle with radius $2$. What is the area of the region shown?

(A) $3\sqrt{3}-\pi$ (B) $4\sqrt{3}-\frac{4\pi}{3}$ (C) $2\sqrt{3}$ (D) $4\sqrt{3}-\frac{2\pi}{3}$ (E) $4+\frac{4\pi}{3}$.

AMC 8, 2016, Problem 2

In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$?

(A) $12$ (B) $15$ (C) $18$ (D) $20$ (E) $24$.

AMC 8, 2016, Problem 22

Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. What is the area of the "bat wings" (shaded area)?

(A) $2$ (B) $2 \frac{1}{2}$ (C) $3$ (D) $3 \frac{1}{2}$ (E) $5$.

AMC 8, 2016, Problem 23

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?

(A) $90$ (B) $105$ (C) $120$ (D) $135$ (E) $150$

AMC 8, 2016, Problem 25

A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

(A) $4 \sqrt{3}$ (B) $\frac{120}{17}$ (C) $10$ (D) $\frac{17\sqrt{2}}{2}$

(E) $\frac{17\sqrt{3}}{2}$.

AMC 8, 2015, Problem 1

How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are $3$ feet in a yard.)

(A) $12$ (B)$36$ (C) $108$ (D) $324$ (E) $972$.

AMC 8, 2015, Problem 2

Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?

(A) $\frac{11}{32}$ (B) $\frac{3}{8}$ (C) $\frac{13}{32}$ (D) $\frac{7}{16}$ (E) $\frac{15}{32}$.

AMC 8, 2015, Problem 6

In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$?

(A)$100$ (B) $420$ (C) $500$ (D) $609$ (E) $701$.

AMC 8, 2015, Problem 8

What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?

(A) $24$ (B) $29$ (C) $43$ (D) $48$ (E) $57$.

AMC 8, 2015, Problem 12

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?

(A) $6$ (B) $12$ (C) $18$ (D) $ 24$ (E) $ 36$.

AMC 8, 2015, Problem 19

A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?

(A) $\frac{1}{6}$ (B)$\frac{1}{5}$ (C) $\frac{1}{4}$ (D) $\frac{1}{3}$ (E) $\frac{1}{2}$

AMC 8, 2015, Problem 21

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?

(A) $6\sqrt{2}$ (B)$9$ (C) $12$ (D) $9\sqrt{2}$ (E) $32$.

AMC 8, 2015, Problem 25

One-inch squares are cut from the corners of this $5$ inch square. What is the area in square inches of the largest square that can fit into the remaining space?

(A) $ 9$ (B) $12\frac{1}{2}$ (C) $15$ (D)$15\frac{1}{2}$ (E)$17$

AMC 8, 2014, Problem 9

In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^{\circ}$. What is the degree measure of $\angle ADB$?

(A) $100$ (B)$120$ (C) $135$ (D) $140$ (E) $150$.

AMC 8, 2014, Problem 14

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?

(A) $12$ (B) $13$ (C) $14$ (D) $15$ (E) $16$.

AMC 8, 2014, Problem 15

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?

(A) $75$ (B) $80$ (C) $90$ (D) $120$ (E) $150$.

AMC 8, 2014, Problem 19

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

(A) $\frac{5}{54}$ (B) $\frac{1}{9}$ (C) $\frac{5}{27}$ (D)$\frac{2}{9}$

(E) $\frac{1}{3}$

AMC 8, 2014, Problem 20

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle with a radius of $1$ is centered at $A$, a circle with a radius of $2$ is centered at $B$, and a circle with a radius of $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

(A) $3.5$ (B) $4.0$ (C) $4.5$ (D) $5.0$ (E) $5.5$.

AMC 8, 2013, Problem 18

Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

(A)$ 204$ (B) $ 280$ (C) $320$ (D) $340$ (E) $600$.

AMC 8, 2013, Problem 20

A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?

(A)$ \frac{\pi}{2}$ (B) $ \frac{2\pi}{3} $ (C)$ \pi $ (D)$ \frac{4\pi}{3}$ (E)$ \frac{5\pi}{3}$.

AMC 8, 2013, Problem 21

Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

(A)$3$ (B) $6$ (C) $ 9$ (D) $ 12$ (E) $18$.

AMC 8, 2013, Problem 23

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?

(A)$ 7$ (B)$ 7.5 $ (C)$8 $ (D)$ 8.5 $ (E)$9$

AMC 8, 2013, Problem 24

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares?

(A)$\frac{1}{4}$ (B)$\frac{7}{24}$ (C)$\frac{1}{3}$ (D)$\frac{3}{8}$ (E)$\frac{5}{12}$.

AMC 8, 2013, Problem 25

A ball with diameter $4$ inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from $A$ to $B$?

(A)$ 238\pi$ (B)$240\pi$ (C)$260\pi$ (D)$280\pi$ (E)$500\pi$.

AMC 8, 2012, Problem 5

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is the length in $X$, in centimeters?

(A)$1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$.

AMC 8, 2012, Problem 6

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches?

(A) $36$ (B) $40$ (C) $64$ (D) $72$ (E) $88$

AMC 8, 2012, Problem 17

A square with integer side length is cut into $10$ squares, all of which have integer side length and at least $8$ of which have area $1$. What is the smallest possible value of the length of the side of the original square?

(A) $3$ (B) $4$ (C) $5$ (D) $6$ (E) $7$

AMC 8, 2012, Problem 21

Marla has a large white cube that has an edge of $10$ feet. She also has enough green paint to cover $300$ square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?

(A) $5\sqrt2$ (B) $10$ (C) $10\sqrt2$ (D) $50$ (E) $50\sqrt2$.

AMC 8, 2012, Problem 23

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is $4$, what is the area of the hexagon?

(A) $4$ (B) $5$ (C) $6$ (D) $4\sqrt3$ (E) $6\sqrt3$.

AMC 8, 2012, Problem 24

A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

(A) $\frac{4-\pi}{\pi}$ (B) $\frac{1}{\pi}$ (C) $\frac{\sqrt2}{\pi}$ (D) $\frac{\pi-1}{\pi}$ (E) $\frac{3}{\pi}$

AMC 8, 2009, Problem 7

The triangular plot of $ACD$ lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land $ACD$?

(A)$ 2$ (B) $3$ (C) $ 4.5$ (D) $6$ (E) $ 9$.

AMC 8, 2009, Problem 9

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

(A)$ 21$ (B)$23$ (C)$25$ (D)$27$ (E)$29$.

AMC 8, 2009, Problem 18

The diagram represents a $7$-foot-by-$7$-foot floor that is tiled with $1$-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a $15$-foot-by-$15$-foot floor is to be tiled in the same manner, how many white tiles will be needed?

(A) $49$ (B) $57$ (C) $64$ (D) $96$ (E) $126$.

AMC 8, 2009, Problem 20

How many non-congruent triangles have vertices at three of the eight points in the array shown below?

(A)$ 5$ (B) $6$ (C) $ 7$ (D) $8$ (E) $ 9$.

AMC 8, 2009, Problem 25

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $1/2$ foot from the top face. The second cut is $1/3$ foot below the first cut, and the third cut is $1/17$ foot below the second cut. From the top to the bottom the pieces are labeled $A, B, C$, and $D$. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?

(A)$6$ (B)$7$ (C)$\frac{419}{51}$ (D)$\frac{158}{17}$ (E)$11$.

AMC 8, 2008, Problem 16

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?

(A) $1 : 6$ (B) $7 : 36$ (C) $1 : 5$ (D) $7 : 30$ (E) $6 : 25$.

AMC 8, 2008, Problem 18

Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?

(A) $10\pi + 201$ (B) $ 10\pi + 30$ (C) $10 \pi +40$ (D) $20\pi + 20$ (E) $20\ pi +40$.

AMC 8, 2008, Problem 21

Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?

(A) $48$ (B) $75$ (C) $151$ (D) $192$ (E) $603$.

AMC 8, 2008, Problem 23

In square $ABCE, A F=2 F E$ and $C D=2 D E$. What is the ratio of the area of $\triangle B F D$ to the area of square $A B C E ?$

(A) $\frac{1}{6}$ (B) $\frac{2}{9}$ (C) $\frac{5}{18}$ (D) $\frac{1}{3}$

(E) $\frac{7}{20}$.

AMC 8, 2008, Problem 25

Margie's winning art design is shown. The smallest circle has radius $2$ inches, with each successive circle's radius increasing by $2$ inches. Which of the following is closest to the percent of the design that is black?

(A) $41.7$ (B) $44$ (C) $45$ (D) $46$ (E) $48$.

AMC 8, 2007, Problem 8

In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD$ = $AB$ = $3$, and $DC$ = $6$. In addition, $E$ is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\triangle BEC$.

(A) $3$ (B) $4.5$ (C) $6$ (D) $9$ (E) $18$

AMC 8, 2007, Problem 12

A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$ equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

(A) $1: 1$ (B) $6: 5$ (C) $3: 2$ (D) $2: 1$ (E) $3: 1$

AMC 8, 2007, Problem 14

The base of isosceles $\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides?

(A) $5$ (B) $8$ (C) $13$ (D) $14$ (E) $18$

AMC 8, 2007, Problem 23

What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?

(A) $ 4$ (B) $6$ (C) $8$ (D) $10$ (E) $12$

AMC 8, 2006, Problem 5

Points $A, B, C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area $60$, what is the area of the smaller square?

(A) $15$ (B)$20$ (C) $24$ (D) $30$ (E) $40$

AMC 8, 2006, Problem 6

The letter $T$ is formed by placing two $2 \times 4$ inch rectangles next to each other, as shown. What is the perimeter of the $T$, in inches?

(A) $12$ (B) $16$ (C) $20$ (D) $22$ (E) $ 24$

AMC 8, 2006, Problem 7

Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to largest radius.

(A) $ X, Y, Z$ (B) $Z, X, Y$ (C) $Y, X, Z$ (D) $Z, Y, X$ (E) $ X, Z, Y$

AMC 8, 2006, Problem 18

A cube with $3$-inch edges is made using $27$ cubes with $1$-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?

(A) $\frac{1}{9}$ (B) $\frac{1}{4}$ (C)$\frac{4}{9}$ (D)$\frac{5}{9}$ (E)$\frac{19}{27}$

AMC 8, 2006, Problem 19

Triangle $ABC$ is an isosceles triangle with $\overline{AB}=\overline{BC}$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, and $\overline{CE}$ is $11$ units long. Triangle $ABD$ is congruent to triangle $ECD$. What is the length of $\overline{BD}$?

(A) $4$ (B) $4.5$ (C) $5$ (D) $5.5$ (E) $6$

AMC 8, 2005, Problem 3

What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $\overline{B D}$ of square $A B C D ?$

(A)$1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$

AMC 8, 2005, Problem 9

In quadrilateral $A B C D,$ sides $\overline{A B}$ and $\overline{B C}$ both have length $10,$ sides $\overline{C D}$ and $\overline{D A}$ both have length $17,$ and the measure of angle $A D C$ is $60^{\circ} .$ What is the length of diagonal $\overline{A C} ?$

(A) $13.5$ (B) $14$ (C) $15.5$ (D) $17$ (E) $18.5$

AMC 8, 2005, Problem 13

The area of polygon $A B C D E F$ is $52$ with $A B=8, B C=9$ and $F A=5$. What is $D E+E F ?$

(A) $47$ (B) $8$ (C) $9$ (D) $10$ (E) $11$

AMC 8, 2005, Problem 19

What is the perimeter of trapezoid $A B C D ?$

(A) $180$ (B) $188$ (C) $196$ (D) $200$ (E) $204$

AMC 8, 2005, Problem 23

Isosceles right triangle $A B C$ encloses a semicircle of area $2 \pi$. The circle has its center $O$ on hypotenuse $\overline{A B}$ and is tangent to sides $\overline{A C}$ and $\overline{B C}$. What is the area of triangle $A B C$ ?

(A) $6$ (B) $8$ (C) $3 \pi$ (D) $10$ (E) $4 \pi$

AMC 8, 2005, Problem 25

A square with side length $2$ and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

(A) $\frac{2}{\sqrt{\pi}}$ (B) $\frac{1+\sqrt{2}}{2}$ (C) $\frac{3}{2}$
(D) $\sqrt{3}$ (E) $\sqrt{\pi}$

AMC 8, 2004, Problem 14

What is the area enclosed by the geoboard quadrilateral below?

(A) $15$ (B) $18 \frac{1}{2}$ (C) $22 \frac{1}{2}$ (D) $27$ (E) $41$.

AMC 8, 2004, Problem 24

In the figure, $A B C D$ is a rectangle and $E F G H$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $\overline{H E}$ and $\overline{F G}$ ?

(A) $6.8$ (B) $7.1$ (C) $7.6$ (D) $7.8$ (E) $8.1$

AMC 8, 2004, Problem 25

Two $4 \times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

(A) $16-4 \pi$ (B) $16-2 \pi$ (C) $28-4 \pi$ (D) $28-2 \pi$ (E) $32-2 \pi$

AMC 8, 2003, Problem 6

Given the areas of the three squares in the figure, what is the area of the interior triangle?

(A) $13$ (B) $30$ (C) $60$ (D) $300$ (E) $1800$

AMC 8, 2003, Problem 8

Bake Sale
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

Art's cookies are trapezoids.

Roger's cookies are rectangles.

Paul's cookies are parallelograms.

Trisha's cookies are triangles.

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Who gets the fewest cookies from one batch of cookie dough?

(A) Art (B) Roger (C) Paul (D) Trisha (E) There is a tie for fewest.

AMC 8, 2003, Problem 21

The area of trapezoid $A B C D$ is $164 \mathrm{~cm}^{2}$. The altitude is $8 \mathrm{~cm}, A B$ is $10 \mathrm{~cm},$ and $C D$ is 17 $\mathrm{cm}$. What is $B C$, in centimeters?

(A) $9$ (B) $10$ (C) $12$ (D) $15$ (E) $20$

AMC 8, 2003, Problem 22

The following figures are composed of squares and circles. Which figure has a shaded region with largest area?

(A) A only (B) B (C) C only (D) both A and B (E) all are equal

AMC 8, 2003, Problem 25

In the figure, the area of square $W X Y Z$ is $25 \mathrm{~cm}^{2}$. The four smaller squares have sides 1 $\mathrm{cm}$ long, either parallel to or coinciding with the sides of the large square. In $\triangle A B C$, $A B=A C,$ and when $\triangle A B C$ is folded over side $\overline{B C}$, point $A$ coincides with $O,$ the center of square $W X Y Z$. What is the area of $\triangle A B C$, in square centimeters?

(A) $\frac{15}{4}$ (B) $\frac{21}{4}$ (C) $\frac{27}{4}$ (D) $\frac{21}{2}$
(E) $\frac{27}{2}$

AMC 8, 2002, Problem 20

The area of triangle $X Y Z$ is $8$ square inches. Points $A$ and $B$ are midpoints of congruent segments $\overline{X Y}$ and $\overline{X Z}$. Altitude $\overline{X C}$ bisects $\overline{Y Z}$. The area (in square inches) of the shaded region is

(A) $1 \frac{1}{2}$ (B) $2$ (C) $2 \frac{1}{2}$ (D) $3$ (E) $3 \frac{1}{2}$

AMC 8, 2001, Problem 11

Points $A, B, C$ and $D$ have these coordinates: $A(3,2), B(3,-2), C(-3,-2)$ and $D(-3,0)$. The area of quadrilateral $A B C D$ is

(A) $12$
(B) $15$
(C) $18$
(D) $21$
(E) $24$

AMC 8, 2001, Problem 16

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

(A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{3}{4}$ (D) $\frac{4}{5}$
(E) $\frac{5}{6}$.

AMC 8, 2000, Problem 6

Figure $A B C D$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded $L$ -shaped region is

(A) $7$ (B) $10$ (C) $12.5$ (D) $14$ (E) $15$.

AMC 8, 2000, Problem 13

In triangle $C A T,$ we have $\angle A C T=\angle A T C$ and $\angle C A T=36^{\circ} .$ If $\overline{T R}$ bisects $\angle A T C$ then $\angle C R T=$

(A) $36^{\circ}$ (B) $54^{\circ}$ (C) $72^{\circ}$ (D) $90^{\circ}$
(E) $108^{\circ}$.

AMC 8, 2000, Problem 15

Triangles $A B C, A D E,$ and $E F G$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{A C}$ and $\overline{A E},$ respectively. If $A B=4,$ what is the perimeter of figure $A B C D E F G ?$

(A) $12$ (B) $13$ (C) $15$ (D) $18$ (E) $21$.

AMC 8, 2000, Problem 19

Three circular arcs of radius $5$ units bound the region shown. Arcs $A B$ and $A D$ are quarter circles, and arc $B C D$ is a semicircle. What is the area, in square units, of the region?

(A) $25$ (B) $10+5 \pi$ (C) $50$ (D) $50+5 \pi$ (E) $25 \pi$.

AMC 8, 2000, Problem 22

A cube has edge length $2$ . Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to

(A) $10$ (B) $15$ (C) $17$ (D) $21$ (E) $25$.

AMC 8, 2000, Problem 24

If $\angle A=20^{\circ}$ and $\angle A F G=\angle A G F$, then $\angle B+\angle D=$

(A) $48^{\circ}$ (B) $60^{\circ}$ (C) $72^{\circ}$ (D) $80^{\circ}$ (E) $90^{\circ}$.

AMC 8, 2000, Problem 25

The area of rectangle $A B C D$ is 72 . If point $A$ and the midpoints of $\overline{B C}$ and $\overline{C D}$ are joined to form a triangle, the area of that triangle is

(A) $21$ (B) $27$ (C) $30$ (D) $36$ (E) $40$.

AMC 8, 1999, Problem 5

A rectangular garden $60$ feet long and $20$ feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

(A) $100$ (B) $200$ (C) $300$ (D) $400$ (E) $500$.

AMC 8, 1999, Problem 14

In trapezoid $A B C D$, the sides $A B$ and $C D$ are equal. The perimeter of $A B C D$ is

(A) $27$ (B) $30$ (C) $32$ (D) $34$ (E) $48$

AMC 8, 1999, Problem 21

The degree measure of angle $A$ is

(A) $20$ (B) $30$ (C) $35$ (D) $40$ (E) $45$.

AMC 8, 1999, Problem 23

Square $A B C D$ has sides of length $3 .$ Segments $C M$ and $C N$ divide the square's area into three equal parts. How long is segment $C M ?$

(A) $\sqrt{10}$ (B) $\sqrt{12}$ (C) $\sqrt{13}$ (D) $\sqrt{14}$ (E) $\sqrt{15}$.

AMC 8, 1999, Problem 25

Points $B, D,$ and $J$ are midpoints of the sides of right triangle $A C G .$ Points $K, E, I$ are midpoints of the sides of triangle $J D G$, etc. If the dividing and shading process is done $100$ times (the first three are shown) and $A C=C G=6$, then the total area of the shaded triangles is nearest

(A) $6$ (B) $7$ (C) $8$ (D) $9$ (E) $10$.

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