AMERICAN MATHEMATICS COMPETITION 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

ANSWER :

(A) 10

PROBLEM 2 :

Consider these two operations:

$$
\begin{aligned}
a \bullet b & =a^2-b^2 \
a \star b & =(a-b)^2
\end{aligned}
$$

What is the output of $(5 \diamond 3) \star 6$ ?
(A) -20
(B) 4
(C) 16
(D) 100
(E) 220

ANSWER :

(D) 100

PROBLEM 3 :

When three positive integers $a, b$, and $c$ are multiplied together, their product is 100 . Suppose $a<b<c$. In how many ways can the numbers be chosen?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

(E) 4

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?

ANSWER :

(E)

PROBLEM 5 :

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned 6 years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is 30 years. How many years older than Bella is Anna?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER :

PROBLEM 6 :

Three positive integers are equally spaced on a number line. The middle number is 15 , and the largest number is 4 times the smallest number. What is the smallest of these three numbers?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

ANSWER :

PROBLEM 7 :

When the World Wide Web first became popular in the 1990 s, download speeds reached a maximum of about 56 kilobits per second.
Approximately how many minutes would the download of a 4.2 -megabyte song have taken at that speed? (Note that there are 8000 kilobits in a megabyte.)
(A) 0.6
(B) 10
(C) 1800
(D) 7200
(E) 36000

ANSWER :

(B) 10

PROBLEM 8 :

What is the value of

$$
\frac{1}{3} \cdot \frac{2}{4} \cdot \frac{3}{5} \cdots \frac{18}{20} \cdot \frac{19}{21} \cdot \frac{20}{22} ?
$$

(A) $\frac{1}{462}$
(B) $\frac{1}{231}$
(C) $\frac{1}{132}$
(D) $\frac{2}{213}$
(E) $\frac{1}{22}$

ANSWER :

(B) $\frac{1}{231}$.

PROBLEM 9 :

A cup of boiling water $\left(212^{\circ} \mathrm{F}\right)$ is placed to cool in a room whose temperature remains constant at $68^{\circ} \mathrm{F}$. Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?
(A) 77
(B) 86
(C) 92
(D) 98
(E) 104

ANSWER :

(B) 86

PROBLEM 10 :

One sunny day, Ling decided to take a hike in the mountains. She left her house at 8 AM , drove at a constant speed of 45 miles per hour, and arrived at the hiking trail at 10 AM . After hiking for 3 hours, Ling drove home at a constant speed of 60 miles per hour. Which of the following graphs best illustrates the distance between Ling's car and her house over the course of her trip?

ANSWER :

(E)

PROBLEM 11 :

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating 3 inches of pasta from the middle of one piece. In the end, he has 10 pieces of pasta whose total length is 17 inches. How long, in inches, was the piece of pasta he started with?
(A) 34
(B) 38
(C) 41
(D) 44
(E) 47

ANSWER :

(D) 44

PROBLEM 12 :

The arrows on the two spinners shown below are spun. Let the number $N$ equal 10 times the number on Spinner A , added to the number on Spinner B. What is the probability that $N$ is a perfect square number?

(A) $\frac{1}{16}$
(B) $\frac{1}{8}$
(C) $\frac{1}{4}$
(D) $\frac{3}{8}$
(E) $\frac{1}{2}$

ANSWER :

(B) $\frac{1}{8}$

PROBLEM 13 :

How many positive integers can fill the blank in the sentence below?
"One positive integer is ____ more than twice another, and the sum of the two numbers is $28 . "$

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

ANSWER :

(D) 9

PROBLEM 14 :

In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together?
(A) 1
(B) 4
(C) 12
(D) 24
(E) 120

ANSWER :

(D) 24

PROBLEM 15 :

Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

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

ANSWER :

PROBLEM 16 :

Four numbers are written in a row. The average of the first two is 21 , the average of the middle two is 26 , and the average of the last two is 30 . What is the average of the first and last of the numbers?
(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

ANSWER :

(B) 25

PROBLEM 17 :

If $n$ is an even positive integer, the double factorial notation $n!!$ represents the product of all the even integers from 2 to $n$. For example, $8!!=2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum?

$$
2!!+4!!+6!!+\cdots+2018!!+2020!!+2022!!
$$

(A) 0
(B) 2
(C) 4
(D) 6
(E) 8

ANSWER :

(B) 2

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

ANSWER :

PROBLEM 19 :

Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores.

Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85 . What is the minimum number of students who received extra points?
(Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

ANSWER :

PROBLEM 20 :

The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$ ?

(A) -1
(B) 5
(C) 6
(D) 8
(E) 9

ANSWER :

(D) 8

PROBLEM 21 :

Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

ANSWER :

PROBLEM 22 :

A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?

(A) 17
(B) 19
(C) 20
(D) 21
(E) 23

ANSWER :

(A) 17

PROBLEM 23 :

A △ or $\bigcirc$ is placed in each of the nine squares in a 3 -by- 3 grid. Shown below is a sample configuration with three $\triangle s$ in a line.

How many configurations will have three △ s in a line and three □ s in a line?
(A) 39
(B) 42
(C) 78
(D) 84
(E) 96

ANSWER :

D) 84

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

ANSWER :

PROBLEM 25 :

A cricket randomly hops between 4 leaves, on each turn hopping to one of the other 3 leaves with equal probability. After 4 hops what is the probability that the cricket has returned to the leaf where it started?

(A) $\frac{2}{9}$
(B) $\frac{19}{80}$
(C) $\frac{20}{81}$
(D) $\frac{1}{4}$
(E) $\frac{7}{27}$

ANSWER :

(E) $\frac{7}{27}$

AMERICAN MATHEMATICS COMPETITION 8 - 2021

PROBLEM 1 :

Which of the following values is largest?
(A) $2+0+1+7$
(B) $2 \times 0+1+7$
(C) $2+0 \times 1+7$
(D) $2+0+1 \times 7$
(E) $2 \times 0 \times 1 \times 7$

ANSWER : (A) $2+0+1+7$

PROBLEM 2 :

Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together?

(A) 70
(B) 84
(C) 100
(D) 106
(E) 120

ANSWER :(E) 120

PROBLEM 3 :

What is the value of the expression $\sqrt{16 \sqrt{8 \sqrt{4}}}$ ?
(A) 4
(B) $4 \sqrt{2}$
(C) 8
(D) $8 \sqrt{2}$
(E) 16

ANSWER : (C) 8

PROBLEM 4 :

When 0.000315 is multiplied by $7,928,564$ the product is closest to which of the following?
(A) 210
(B) 240
(C) 2100
(D) 2400
(E) 24000

ANSWER : (D) 2400

PROBLEM 5 :

What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$ ?
(A) 1020
(B) 1120
(C) 1220
(D) 2240
(E) 3360

ANSWER : (B) 1120

PROBLEM 6 :

If the degree measures of the angles of a triangle are in the ratio $3: 3: 4$, what is the degree measure of the largest angle of the triangle?
(A) 18
(B) 36
(C) 60
(D) 72
(E) 90

ANSWER : (D) 72

PROBLEM 7 :

Let $\boldsymbol{Z}$ be a 6 -digit positive integer, such as 247247 , whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$ ?
(A) 11
(B) 19
(C) 101
(D) 111
(E) 1111

ANSWER : (A) 11

PROBLEM 8 :

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.

(2) It is even.
(3) It is divisible by 7 .
(4) One of its digits is 9 .

This information allows Malcolm to determine Isabella's house number. What is its units digit?
(A) 4
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER : (D) 8

PROBLEM 9 :

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER : (D) 4

PROBLEM 10 :

A box contains five cards, numbered $1,2,3,4$, and 5 . Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
(A) $\frac{1}{10}$
(B) $\frac{1}{5}$
(C) $\frac{3}{10}$
(D) $\frac{2}{5}$
(E) $\frac{1}{2}$

ANSWER : (D) $\frac{2}{5}$

PROBLEM 11 :

A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37 , how many tiles cover the floor?
(A) 148
(B) 324
(C) 361
(D) 1296
(E) 1369

ANSWER : (C) 361

PROBLEM 12 :

The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4,5 , and 6 lies between which of the following pairs of numbers?
(A) 2 and 19
(B) 20 and 39
(C) 40 and 59
(D) 60 and 79
(E) 80 and 124

ANSWER : (D) 60 and 79

PROBLEM 13 :

Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER : (B) 1

PROBLEM 14 :

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80 \%$ of the problems she solved alone, but overall $88 \%$ of her answers were correct. Zoe had correct answers to $90 \%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers?
(A) 89
(B) 92
(C) 93
(D) 96
(E) 98

ANSWER : (C) 93

PROBLEM 15 :

In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.

(A) 8
(B) 9
(C) 12
(D) 24
(E) 36

ANSWER : (D) 24

PROBLEM 16 :

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

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

ANSWER : (D) $\frac{12}{5}$

PROBLEM 17 :

Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have?
(A) 9
(B) 27
(C) 45
(D) 63
(E) 81

ANSWER : (C) 45

PROBLEM 18 :

In the non-convex quadrilateral $A B C D$ shown below, $\angle B C D$ is a right angle, $A B=12, B C=4, C D=3$, and $A D=13$.

What is the area of quadrilateral $A B C D$ ?
(A) 12
(B) 24
(C) 26
(D) 30
(E) 36

ANSWER : (B) 24

PROBLEM 19 :

For any positive integer $M$, the notation $M!$ denotes the product of the integers 1 through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?
(A) 23
(B) 24
(C) 25
(D) 26
(E) 27

ANSWER : (D) 26

PROBLEM 20 :

An integer between 1000 and 9999 , inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
(A) $\frac{14}{75}$
(B) $\frac{56}{225}$
(C) $\frac{107}{400}$
(D) $\frac{7}{25}$
(E) $\frac{9}{25}$

ANSWER : (B) $\frac{56}{225}$

PROBLEM 21 :

Suppose $a, b$, and are nonzero real numbers, and . What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b c}{|a b c|}$ ?
(A) 0
(B) 1 and - 1
(C) 2 and - 2
(D) 0,2, and - 2
(E) 0 , 1 , and -1

ANSWER : (A) 0

PROBLEM 22 :

In the right triangle $A B C, A C=12, B C=5$, 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) $\frac{60}{13}$

ANSWER : (D) $\frac{10}{3}$

PROBLEM 23 :

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
(A) 10
(B) 15
(C) 25
(D) 50
(E) 82

ANSWER : (C) 25

PROBLEM 24 :

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2021. On how many days during the next year did she not receive a phone call from any of her grandchildren?
(A) 78
(B) 80
(C) 144
(D) 146
(E) 152

ANSWER : (D) 146

PROBLEM 25 :

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

ANSWER : (B)

AMERICAN MATHEMATICS COMPETITION - 2020

Problem 1 :
Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?
(A) 6
(B) 8
(C) 12
(D) 18
(E) 24

ANSWER :

(E) 24

Problem 2 :
Four friends do yardwork for their neighbors over the weekend, earning $\$ 15, \$ 20, \$ 25$, and $\$ 40$, respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $\$ 40$ give to the others?
(A) $\$ 5$
(B) $\$ 10$
(C) $\$ 15$
(D) $\$ 20$
(E) $\$ 25$

ANSWER :

Problem 3 :
Carrie has a rectangular garden that measures 6 feet by 8 feet. She plants the entire garden with strawberry plants. Carrie is able to plant 4 strawberry plants per square foot, and she harvests an average of 10 strawberries per plant. How many strawberries can she expect to harvest?
(A) 560
(B) 960
(C) 1120
(D) 1920
(E) 3840

ANSWER :

(D) 1920

Problem 4 :
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?

(A) 35
(B) 37
(C) 39
(D) 43
(E) 49

ANSWER :

(B) 37

Problem 5 :
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25

ANSWER :

Problem 6 :
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
(A) Aaron
(B) Darren
(C) Karen
(D) Maren
(E) Sharon

ANSWER :

(A) Aaron

Problem 7 :
How many integers between 2020 and 2400 have four distinct digits arranged in increasing order? (For example, 2357 is one such integer.)
(A) 9
(B) 10
(C) 15
(D) 21
(E) 28

ANSWER :

Problem 8 :

Ricardo has 2020 coins, some of which are pennies (1-cent coins) and the rest of which are nickels (5-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
(A) 8062
(B) 8068
(C) 8072
(D) 8076
(E) 8082

ANSWER :

Problem 9 :
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

(A) 12
(B) 16
(C) 18
(D) 20
(E) 24

ANSWER :

(D) 20

Problem 10 :
Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
(A) 6
(B) 8
(C) 12
(D) 18
(E) 24

ANSWER :

Problem 11 :
After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

(A) 6
(B) 12
(C) 18
(D) 20
(E) 24

ANSWER :

(E) 24

Problem 12 :
For positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to 1 . (For example, $6!=6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.) What value of $N$ satisfies the following equation?

$$
5!\cdot 9!=12 \cdot N!
$$

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

ANSWER :

(A) 10

Problem 13 :
Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a $60 \%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
(A) 6
(B) 9
(C) 12
(D) 18
(E) 24

ANSWER :

(B) 9

Problem 14 :
There are 20 cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all 20 cities?

(A) 65,000
(B) 75,000
(C) 85,000
(D) 95,000
(E) 105,000

ANSWER :

(D) 95,000

Problem 15 :
Suppose $15 \%$ of $x$ equals $20 \%$ of $y$. What percentage of $x$ is $y$ ?

(A) 5
(B) 35
(C) 75
(D) $133 \frac{1}{3}$
(E) 300

ANSWER :

Problem 16 :
Each of the points $A, B, C, D, E$, and $F$ in the figure below represents a different digit from 1 to 6 . Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47 . What is the digit represented by $B$ ?

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

ANSWER :

(E) 5

Problem 17 :
How many factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1 , $2,3,4,6$, and 12 .)
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

ANSWER :

(B) 7

Problem 18 :
Rectangle $A B C D$ is inscribed in a semicircle with diameter $\overline{F E}$, as shown in the figure. Let $D A=$ 16, and let $F D=A E=9$. What is the area of $A B C D$ ?

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

ANSWER :

(A) 240

Problem 19 :
A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15 ?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

ANSWER :

(B) 4

Problem 20 :
A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

(A) 22.2
(B) 24.2
(C) 33.2
(D) 35.2
(E) 37.2

ANSWER :

(B) 24.2

Problem 21 :
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7 -step paths are there from $P$ to $Q$ ? (The figure shows a sample path.)

(A) 28
(B) 30
(C) 32
(D) 33
(E) 35

ANSWER :

(A) 28

Problem 22 :
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7$, the machine will output $3 \cdot 7+1=22$. Then if the output is repeatedly inserted into the machine five more times, the final output is 26 .

$$
7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26
$$

When the same 6 -step process is applied to a different starting value of $N$, the final output is 1 . What is the sum of all such integers $N$ ?

$$
N \rightarrow {-} \rightarrow {-} \rightarrow \rightarrow {-} \rightarrow {-} \rightarrow 1
$$

(A) 73
(B) 74
(C) 75
(D) 82
(E) 83

ANSWER :

(E) 83

Problem 23 :
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
(A) 120
(B) 150
(C) 180
(D) 210
(E) 240

ANSWER :

(B) 150

Problem 24 :
A large square region is paved with $n^{2}$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n=3$. When $n=$ 24 , the 576 gray tiles cover $64 \%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$ ?

(A) $\frac{6}{25}$
(B) $\frac{1}{4}$
(C) $\frac{9}{25}$
(D) $\frac{7}{16}$
(E) $\frac{9}{16}$

ANSWER :

(A) $\frac{6}{25}$

Problem 25 :
Rectangles $R_{1}$ and $R_{2}$, and squares $S_{1}, S_{2}$, and $S_{3}$, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_{2}$ in units?

(A) 651
(B) 655
(C) 656
(D) 662
(E) 666

ANSWER :

(A) 651

American Mathematics Competition - 2019

Question 1 :

Ike and Mike go into a sandwich shop with a total of $\$ 30.00$ to spend. Sandwiches cost $\$ 4.50$ each and soft drinks cost $\$ 1.00$ each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Answer 1 :

(D) 9

Question 2:

Three identical rectangles are put together to form rectangle $A B C D$, 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 $A B C D$ ?

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

Answer 2 :

(E) 150

Question 3 :

Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest?
(A) $\frac{15}{11}<\frac{17}{13}<\frac{19}{15}$
(B) $\frac{15}{11}<\frac{19}{15}<\frac{17}{13}$
(C) $\frac{17}{13}<\frac{19}{15}<\frac{15}{11}$
(D) $\frac{19}{15}<\frac{15}{11}<\frac{17}{13}$
(E) $\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

Answer 3 :

(E) $\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

Question 4 :

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

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

Answer 4 :

(D) 120

Question 5 :

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $l$ from start to finish?

Answer 5 :

Question 6 :

There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other 80 points, what is the probability that the line $P Q$ is a line of symmetry for the square?

(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{2}{5}$
(D) $\frac{9}{20}$
(E) $\frac{1}{2}$

Answer 6 :

Question 7 :

Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76,94 , and 87 . In order to average 81 for all five tests, what is the lowest score she could earn on one of the other two tests?
(A) 48
(B) 52
(C) 66
(D) 70
(E) 74

Answer 7 :

(A) 48

Question 8 :

Gilda has a bag of marbles. She gives $20 \%$ of them to her friend Pedro. Then Gilda gives $10 \%$ of what is left to another friend, Ebony. Finally, Gilda gives $25 \%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
(A) 20
(B) $33 \frac{1}{3}$
(C) 38
(D) 45
(E) 54

Answer 8 :

(E) 54

Question 9 :

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?
(A) $1: 4$
(B) $1: 2$
(C) $1: 1$
(D) $2: 1$
(E) $4: 1$

Answer 9 :

(B) $1: 2$

Question 10 :

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually 21 participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

(A)The mean increases by 1 and the median does not change.
(B)The mean increases by 1 and the median increases by 1 .
(C) The mean increases by 1 and the median increases by 5 .
(D)The mean increases by 5 and the median increases by 1 .
(E)The mean increase by 5 and the median increases by 5 .

Answer 10 :

(B)The mean increases by 1 and the median increases by 1 .

Question 11 :

The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
(A) 16
(B) 53
(C) 31
(D) 39
(E) 70

Answer 11 :

(D) 39

Question 12 :

The faces of a cube are painted in six different colors: red $(R)$, white $(W)$, green $(G)$, brown $(B)$, aqua $(A)$, and purple $(P)$. Three views of the cube are shown below. What is the color of the face opposite the aqua face?

Answer 12 :

(A) R

Question 13 :

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$ ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer 13 :

(A) 2

Question 14 :

Isabella has 6 coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
(A) Monday
(B) Tuesday
(C) Wednesday
(D) Thursday
(E) Friday

Answer 14 :

Question 15 :

On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
(A) $\frac{14}{85}$
(B) $\frac{7}{25}$
(C) $\frac{2}{5}$
(D) $\frac{4}{7}$
(E) $\frac{7}{10}$

Answer 15 :

(B) $\frac{7}{25}$

Question 16 :

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
(A) 45
(B) 62
(C) 90
(D) 110
(E) 135

Answer 16 :

(D) 110

Question 17 :

What is the value of the product

$$
\left(\frac{1 \cdot 3}{2 \cdot 2}\right)\left(\frac{2 \cdot 4}{3 \cdot 3}\right)\left(\frac{3 \cdot 5}{4 \cdot 4}\right) \cdots\left(\frac{97 \cdot 99}{98 \cdot 98}\right)\left(\frac{98 \cdot 100}{99 \cdot 99}\right) ?
$$

(A) $\frac{1}{2}$
(B) $\frac{50}{99}$
(C) $\frac{9800}{9801}$
(D) $\frac{100}{99}$
(E) 50

Answer 17 :

(B) $\frac{50}{99}$

Question 18 :

The faces of each of two fair dice are numbered $1,2,3,5,7$, and 8 . When the two dice are tossed, what is the probability that their sum will be an even number?
(A) $\frac{4}{9}$
(B) $\frac{1}{2}$
(C) $\frac{5}{9}$
(D) $\frac{3}{5}$
(E) $\frac{2}{3}$

Answer 18 :

Question 19 :

In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
(A) 22
(B) 23
(C) 24
(D) 26
(E) 30

Answer 19 :

Question 20 :

How many different real numbers $x$ satisfy the equation

$$
\left(x^2-5\right)^2=16 ?
$$

(A) 0
(B) 1
(C) 2
(D) 4
(E) 8

Answer 20 :

(D) 4

Question 21 :

What is the area of the triangle formed by the lines $y=5, y=1+x$, and $y=1-x$ ?
(A) 4
(B) 8
(C) 10
(D) 12
(E) 16

Answer 21 :

(E) 16

Question 22 :

A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84 \%$ of the original price, by what percent was the price increased and decreased?
(A) 16
(B) 20
(C) 28
(D) 36
(E) 40

Answer 22 :

(E) 40

Question 23 :

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

Answer 23 :

(B) 11

Question 24 :

In triangle $\triangle A B C$, point $D$ divides side $\overline{A C}$ so that $A D: D C=1: 2$. Let $E$ be the midpoint of $\overline{B D}$ and let $F$ be the point of intersection of line $\overline{B C}$ and line $\overline{A E}$. Given that the area of $\triangle A B C$ is 360, what is the area of $\triangle E B F$ ?

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

Answer 24 :

(B) 30

Question 25 :

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
(A) 105
(B) 114
(C) 190
(D) 210
(E) 380

Answer 25 :

American Mathematics Competition 8 - 2013

Problem 1

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars.

What is the smallest number of additional cars she must buy in order to be able to arrange her cars in this way?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer :

(A) 1

Problem 2

A sign at the fish market says, " $50 \%$ off, today only: half-pound packages for just $\$ 3$ per package."

What is the regular price for a full pound of fish, in dollars?
(A) 6
(B) 9
(C) 10
(D) 12
(E) 15

Answer:

(D) 12

Problem 3

What is the value of $4 \cdot(-1+2-3+4-5+6-7+\cdots+1000)$ ?
(A) -10
(B) 0
(C) 1
(D) 500
(E) 2000

Answer:

(E) 2000

Problem 4

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money,

each of her seven friends paid an extra $\$ 2.50$ to cover her portion of the total bill. What was the total bill?
(A) $\$ 120
(B) $\$ 128
(C) $\$ 140
(D) $\$ 144
(E) $\$ 160

Answer:

Problem 5

Hammie is in the $6^{\text {th }}$ grade and weighs 106 pounds.

His quadruplet sisters are tiny babies and weigh $5,5,6$, and 8 pounds.

Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
(A) median, by 60
(B) median, by 20
(C) average, by 5
(D) average, by 15
(E) average, by 20

Answer:

(E) average, by 20

Problem 6

The number in each box below is the product of the numbers in the two boxes that touch it in the row above.

For example, $30=6 \times 5$. What is the missing number in the top row?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer:

Peoblem 7

Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass,

Trey counted 6 cars in the first 10 seconds.

It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed.

Which of the following was the most likely number of cars in the train?
(A) 60
(B) 80
(C) 100
(D) 120
(E) 140

Answer:

Problem 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
(A) 1/8
(B) 1/4
(C) 3/8
(D) 1/2
(E) 3/4

Answer:

Problem 9
The Incredible Hulk can double the distance he jumps with each succeeding jump.

If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on,

then on which jump will he first be able to jump more than 1 kilometer?
(A) 9th
(B) 10th
(C) 11th
(D) 12th
(E) 13th

Answer:

Problem 10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
(A) 110
(B) 165
(C) 330
(D) 625
(E) 660

Answer:

Problem 11
Ted's grandfather used his treadmill on 3 days this week.

He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour.

He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday.

If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill.

How many minutes less?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

(D) 4

Problem 12
At the 2013 Winnebago Country Fair a vendor is offering a "fair special" on sandals.

If you buy one pair of sandals at the regular price of $\$ 50$, you get a second pair at a $40 \%$ discount,

and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals.

What percentage of the $\$ 150$ regular price did he save?
(A) 25
(B) 30
(C) 33
(D) 40
(E) 45

Answer:

(B) 30

Problem 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score.

By which of the following might her incorrect sum have differed from the correct one?
(A) 45
(B) 46
(C) 47
(D) 48
(E) 49

Answer:

(A) 45

Problem 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand.

Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
(A) 1/4
(B) 1/3
(C) 3/8
(D) 1/2
(E) 2/3

Answer:

Problem 15
If $3^{p}+3^{4}=90,2^{r}+44=76$, and $5^{3}+6^{s}=1421$, what is the product of $p, r$, and $s$ ?
(A) 27
(B) 40
(C) 50
(D) 70
(E) 90

Answer:

(B) 40

Problem 16
A number of students from Fibonacci Middle School are taking part in a community service project.

The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $5: 3$, and the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$-graders is $8: 5$.

What is the smallest number of students that could be participating in the project?
(A) 16
(B) 40
(C) 55
(D) 79
(E) 89

Answer:

(E) 89

Problem 17
The sum of six consecutive positive integers is 2013 . What is the largest of these six integers?
(A) 335
(B) 338
(C) 340
(D) 345
(E) 350

Answer:

(B) 338

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

Answer:

(B) 280

Problem 19
Bridget, Cassie, and Hannah are discussing the results of their last math test.

Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show their tests to anyone.

Cassie says, "I didn't get the lowest score in our class," and Bridget adds, "I didn't get the highest score.

" What is the ranking of the three girls from highest to lowest?
(A) Hannah, Cassie, Bridget
(B) Hannah, Bridget, Cassie
(C) Cassie, Bridget, Hannah
(D) Cassie, Hannah, Bridget
(E) Bridget, Cassie, Hannah

Answer:

(D) Cassie, Hannah, Bridget

Problem 20
A $1 \times 2$ rectangle is inscribed in a semicircle with the 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) $\frac{5 \pi}{3}$

Answer:

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 of City Park, 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

Answer:

(E) 18

Problem 22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks high.

How many toothpicks are used altogether?
(A) 1920
(B) 1952
(C) 1980
(D) 2013
(E) 3932

Answer:

(E) 3932

Problem 23
Angle $A B C$ of $\triangle A B C$ is a right angle.

The sides of $\triangle A B C$ are the diameters of semicircles as shown.

The area of the semicircle on $\overline{A B}$ equals $8 \pi$,

and the arc of the semicircle on $\overline{A C}$ has length $8.5 \pi$.

What is the radius of the semicircle on $\overline{B C}$ ?
(A) 7
(B) 7.5
(C) 8
(D) 8.5
(E) 9

Answer:

(B) 7.5

Problem 24
Squares $A B C D, E F G H$, and $G H I J$ are equal in area. Points $C$ and $D$ are the midpoints of sides $I H$ and $H E$, respectively. What is the ratio of the area of the shaded pentagon $A J I C B$ 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) $\frac{5}{12}$

Answer:

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 in inches 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) $500 \pi$

Answer:

(A) $238 \pi$

American Mathematics Competition 8 - 2008

Problem 1

Susan had $\$ 50$ to spend at the carnival. She spent $\$ 12$ on food and twice as much on rides. How many dollars did she have left to spend?
(A) 12
(B) 14
(C) 26
(D) 38
(E) 50

Answer : B

Problem 2

The ten-letter code BEST OF LUCK represents the ten digits $0-9$, in order. What 4 -digit number is represented by the code word CLUE?
(A) 8671
(B) 8672
(C) 9781
(D) 9782
(E) 9872

Answer : A

Problem 3

If February is a month that contains Friday the $13^{\text {th }}$, what day of the week is February 1?
(A) Sunday
(B) Monday
(C) Wednesday
(D) Thursday
(E) Saturday

Answer : A

Problem 4

In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1 , and the three trapezoids are congruent. What is the area of one of the trapezoids?

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

Answer : C

Problem 5

Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661. What was his average speed in miles per hour?
(A) 15
(B) 16
(C) 18
(D) 20
(E) 22

Answer : E

Problem 6
In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
(A) $3: 10$
(B) $3: 8$
(C) $3: 7$
(D) $3: 5$
(E) $1: 1$

Answer : D

Problem 7

If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$ ?
(A) 27
(B) 29
(C) 45
(D) 105
(E) 127

Answer : E

Problem 8
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
(A) 60
(B) 70
(C) 75
(D) 80
(E) 85

Answer : D

Problem 9
In 2005 Tycoon Tammy invested Dollar $100$ for two years. During the the first year her investment suffered a $15 Dollar $ loss, but during the second year the remaining investment showed a $20 $ Dollar gain. Over the two-year period, what was the change in Tammy's investment?
(A) 5 Dollar loss
(B) 2 Dollar loss
(C) 1 Dollar gain
(D) 2 Dollar gain
(E) 5 Dollar gain

Answer: D

Problem 10
The average age of the 6 people in Room A is 40 . The average age of the 4 people in Room B is 25. If the two groups are combined, what is the average age of all the people?
(A) 32.5
(B) 33
(C) 33.5
(D) 34
(E) 35

Answer : D

11. Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
(A) 7
(B) 13
(C) 19
(D) 39
(E) 46

Answer : A

Problem 12
A ball is dropped from a height of 3 meters. On its first bounce it rises to a height of 2 meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of 0.5 meters?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer : C

Problem 13
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122,125 and 127 pounds. What is the combined weight in pounds of the three boxes?
(A) 160
(B) 170
(C) 187
(D) 195
(E) 354

Answer : C

Problem 14
Three A's, three B's, and three C's are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer : C

Problem 15

In Theresa's first 8 basketball games, she scored $7,4,3,6,8,3,1$ and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
(A) 35
(B) 40
(C) 48
(D) 56
(E) 72

Answer : B

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) $6: 25$

Answer : D

Problem 17

Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
(A) 76
(B) 120
(C) 128
(D) 132
(E) 136

Answer : D

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

Answer : E

Problem 19

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?

(A) $\frac{1}{4}$
(B) $\frac{2}{7}$
(C) $\frac{4}{11}$
(D) $\frac{1}{2}$
(E) $\frac{4}{7}$

Answer : B

Problem 20

The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
(A) 12
(B) 17
(C) 24
(D) 27
(E) 36

Answer : B

Problem 21

Jerry cuts a wedge from a $6-\mathrm{cm}$ 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

Answer : C

Problem 22

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3 n$ three-digit whole numbers?
(A) 12
(B) 21
(C) 27
(D) 33
(E) 34

Answer : A

Problem 23
In square $A B C E, A F=2 F E$ 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) $\frac{7}{20}$

Answer : C

Problem 24
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
(A) $\frac{1}{10}$
(B) $\frac{1}{6}$
(C) $\frac{11}{60}$
(D) $\frac{1}{5}$
(E) $\frac{7}{30}$

Answer : C

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. Approximately what percent of the design is black?

(A) 42
(B) 44
(C) 45
(D) 46
(E) 48

Answer : A

AMERICAN MATHEMATICS COMPETITION 8 - 2014

Problem 1

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$ ?
(A) -10
(B) -6
(C) 0
(D) 6
(E) 10

Answer

(A) -10

Problem 2

Paul owes Paula 35 cents and has a pocket full of 5 -cent coins, 10 -cent coins, and 25 -cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer

(E) 5

Problem 3

Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
(A) 240
(B) 250
(C) 260
(D) 270
(E) 280

Answer

(B) 250 pages.

Problem 4

The sum of two prime numbers is 85 . What is the product of these two prime numbers?
(A) 85
(B) 91
(C) 115
(D) 133
(E) 166

Answer

(E) 166.

Problem 5

Margie's car can go 32 miles on a gallon of gas, and gas currently costs $\$ 4$ per gallon. How many miles can Margie drive on $\$ 20$ worth of gas?
(A) 64
(B) 128
(C) 160
(D) 320
(E) 640

Answer

Problem 6

Six rectangles each with a common base width of 2 have lengths of $1,4,9,16,25$, and 36 . What is the sum of the areas of the six rectangles?
(A) 91
(B) 93
(C) 162
(D) 182
(E) 202

Answer

(D) 182.

Problem 7

There are four more girls than boys in Ms. Raub's class of 28 students. What is the ratio of number of girls to the number of boys in her class?
(A) $3: 4$
(B) $4: 3$
(C) $3: 2$
(D) $7: 4$
(E) $2: 1$

Answer

(B) $4: 3$

Problem 8

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\$ \underline{1 A 2}$. What is the missing digit $A$ of this 3 -digit number?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer

(D) 3 $\sim$ fn106068.

Problem 9

Answer

(D) 140.

Problem 10

The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8 . In what year was Samantha born?
(A) 1979
(B) 1980
(C) 1981
(D) 1982
(E) 1983

Answer

(A) $1979 \sim$ SweetMango77
corrections made by DrDominic.

Problem 11

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
(A) 4
(B) 5
(C) 6
(D) 8
(E) 10

Answer

(A) 4 is the correct answer.

Problem 12

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?

Answer

(B) Is the correct answer.

Problem 13

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

$\textbf{(A) }$ $n$ and $m$ are even $\qquad\textbf{(B) }$ $n$ and $m$ are odd $\qquad\textbf{(C) }$ $n+m$ is even $\qquad\textbf{(D) }$ $n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

Answer

(D) is odd.

Problem 14

Rectangle

and right triangle

have the same area. They are joined to form a trapezoid, as shown. What is

Answer

(B) Is the correct answer.

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

(D) 120

(E) 150

Answer

Problem 16

The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against nonconference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

(A) 60

(B) 88

(D) 144

(E) 160

Answer

(B) 88.

Problem 17

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?

(A) 4

(B) 6

(D) 10

(E) 12

Answer

(B) 6.

Problem 18

Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?

$(\mathbf{A})$ all 4 are boys $(\mathbf{B})$ all 4 are girls $(\mathbf{C})_{2}$ are girls and 2 are boys $(\mathbf{D})_{3}$ are of one gender and 1 is of the other gender $\boldsymbol{(} \mathbf{E} \boldsymbol{)}$ all of these outcomes are equally likely

Answer

(D) Is the correct answer.

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) $\frac{1}{9}$

(D) $\frac{2}{9}$

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

Answer

(A) $\frac{5}{54}$

Problem 20

Rectangle ABCD has sides $\mathrm{CD}=3$ and $\mathrm{DA}=5$. A circle of radius 1 is centered at A , a circle of radius 2 is centered at B , and a circle of radius 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

(D) 5.0

(E) 5.5

Answer

(B) 4.0

Problem 21

The 7-digit numbers $\underline{74 A 52 B 1}$ and $\underline{326 A B 4 C}$ are each multiples of 3 . Which of the following could be the value of $C$ ?

(A) 1

(B) 2

(D) 5

(E) 8

Answer

(A) 1.

Problem 22

A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?

(A) 1

(B) 3

(D) 7

(E) 9

Answer

(E) 9.

Problem 23

Three members of the Euclid Middle School girls' softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all 2 -digit primes.

Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today's date.

What number does Caitlin wear?

(A) 11

(B) 13

(D) 19

(E) 23

Answer

(A) 11.

Problem 24

One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

(A) 2.5

(B) 3.0

(D) 4.0

(E) 4.5

Answer

Problem 25

A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?

Note: 1 mile= 5280 feet

(A) $\frac{\pi}{11}$

(B) $\frac{\pi}{10}$

(D) $\frac{2 \pi}{5}$

(E) $\frac{2 \pi}{3}$

Answer

(B) $\frac{\pi}{10}$

American Mathematics Competition 8 - 2009

Problem 1

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
(A) 3
(B) 4
(C) 7
(D) 11
(E) 14

Answer: The answer is (E) 14

Problem 2

On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
(A) 7
(B) 32
(C) 35
(D) 49
(E) 112

Answer: $($ D $) 49$

Problem 3

The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7

Answer : (C) 6

Problem 4

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?

Answer : The answer is (B)

Problem 5

A sequence of numbers starts with 1,2 , and 3 . The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
(A) 11
(B) 20
(C) 37
(D) 68
(E) 99

Answer : D

Problem 6

Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?
(A) 40
(B) 42
(C) 44
(D) 46
(E) 48

Answer : A

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

Answer : C

Problem 8

The length of a rectangle is increased by $10 \%$ and the width is decreased by $10 \%$. What percent of the old area is the new area?
(A) 90
(B) 99
(C) 100
(D) 101
(E) 110

Answer : B

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?

Answer : B

Problem 10

On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?

Answer : D

Problem 11

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $\$ 1.43$. Some of the 30 sixth graders each bought a pencil, and they paid a total of $\$ 1.95$. How many more sixth graders than seventh graders bought a pencil?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer : D

Problem 12

The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?

(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{7}{9}$
(E) $\frac{5}{6}$

Answer : D

Problem 13

A three-digit integer contains one of each of the digits 1,3 , and 5 . What is the probability that the integer is divisible by 5 ?
(A) $\frac{1}{6}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{5}{6}$

Answer : B

Problem 14

Austin and Temple are 50 miles apart along Interstate 35 . Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
(A) 46
(B) 48
(C) 50
(D) 52
(E) 54

Answer : B

Problem 15

A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, $1 / 4$ cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?
(A) $5 \frac{1}{8}$
(B) $6 \frac{1}{4}$
(C) $7 \frac{1}{2}$
(D) $8 \frac{3}{4}$
(E) $9 \frac{7}{8}$

Answer : D

Problem 16

How many 3 -digit positive integers have digits whose product equals 24 ?
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

Answer : D

Problem 17

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of 360 and $x$ is a square and the product of 360 and $y$ is a cube. What is the sum of $x$ and $y$ ?
(A) 80
(B) 85
(C) 115
(D) 165
(E) 610

Answer : B

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

Answer : C

Problem 19

Two angles of an isosceles triangle measure $70^{\circ}$ and $x^{\circ}$. What is the sum of the three possible values of $x$ ?
(A) 95
(B) 125
(C) 140
(D) 165
(E) 180

Answer : D

Problem 20

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

Answer : D

Problem 21

Andy and Bethany have a rectangular array of numbers greater than 0 with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is $A$. Bethany adds the numbers in each column. The average of her 75 sums is $B$. What is the value of $\frac{A}{B}$ ?
(A) $\frac{64}{225}$
(B) $\frac{8}{15}$
(C) 1
(D) $\frac{15}{8}$
(E) $\frac{225}{64}$

Answer : D

Problem 22

How many whole numbers between 1 and 1000 do not contain the digit 1 ?
(A) 512
(B) 648
(C) 720
(D) 728
(E) 800

Answer : D

Problem 23

On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
(A) 26
(B) 28
(C) 30
(D) 32
(E) 34

Answer : B

Problem 24

The letters $A, B, C$ and $D$ represent digits. If $AB+CA=DA$ and $AB-C A=A $, what digit does $D$ represent?

Answer : E

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

Answer : E

AMERICAN MATHEMATICS COMPETITION 8 - 2023

PROBLEM 1 :

What is the value of $(8 \times 4+2)-(8+4 \times 2)$ ?
(A) 0
(B) 6
(C) 10
(D) 18
(E) 24

ANSWER :

(D) 18

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?

ANSWER :

(E)

PROBLEM 3 :

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation

$$
(\text { wind chill })=(\text { air temperature })-0.7 \times(\text { wind speed }),
$$

where temperature is measured in degrees Fahrenheit ( ${ }^{\circ} \mathrm{F}$ ) and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ} \mathrm{F}$ and the wind speed is 18 mph . Which of the following is closest to the approximate wind chill?
(A) 18
(B) 23
(C) 28
(D) 32
(E) 35

ANSWER :

(B) 23

PROBLEM 4 :

The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7 . How many of these four numbers are prime?

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

ANSWER :

(D) 3

PROBLEM 5 :

A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
(A) 1250
(B) 1500
(C) 1750
(D) 1800
(E) 2000

ANSWER :

(B) 1500

PROBLEM 6 :

The digits $2,0,2$, and 3 are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

(A) 0
(B) 8
(C) 9
(D) 16
(E) 18

ANSWER :

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

ANSWER :

(B) 1

PROBLEM 8 :

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo's win-loss record?

(A) 000101
(B) 001001
(C) 010000
(D) 010101
(E) 011000

SOLUTION :

(A) 000101

PROBLEM 9 :

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between 4 and 7 meters?

(A) 6
(B) 8
(C) 10
(D) 12
(E) 14

ANSWER :

(B) 8

PROBLEM 10 :

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?
(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{5}{12}$

ANSWER :

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

PROBLEM 11 :

NASA's Perseverance Rover was launched on July 30 , 2020. After traveling $292,526,838$ miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
(A) 6,000
(B) 12,000
(C) 60,000
(D) 120,000
(E) 600,000

ANSWER :

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) $\frac{5}{9}$

ANSWER :

(B) $\frac{11}{36}$

PROBLEM 13 :

Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the start and finish lines. The 3 rd water station is located 2 miles after the 1 st repair station. How long is the race in miles?

(A) 8
(B) 16
(C) 24
(D) 48
(E) 96

ANSWER :

(D) 48

PROBLEM 14 :

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5 -cent, 10 -cent, and 25 -cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$ 7.10$ in postage? (Note: The amount $\$ 7.10$ corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
(A) 45
(B) 46
(C) 51
(D) 54
(E) 55

ANSWER :

(E) 55

PROBLEM 15 :

Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? Here's a hint… if you aren't correct, think about using conversions, maybe that's why you're wrong! -RyanZ4552

(A) 4
(B) 4.2
(C) 4.5
(D) 4.8
(E) 5

ANSWER :

(B) 4.2

PROBLEM 16 :

The letters $\mathrm{P}, \mathrm{Q}$, and R are entered into a $20 \times 20$ table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?

(A) 132 Ps, $134 \mathrm{Qs}, 134 \mathrm{Rs}$
(B) $133 \mathrm{Ps}, 133 \mathrm{Qs}, 134 \mathrm{Rs}$
(C) $133 \mathrm{Ps}, 134 \mathrm{Qs}, 133 \mathrm{Rs}$
(D) $134 \mathrm{Ps}, 132 \mathrm{Qs}, 134 \mathrm{Rs}$
(E) $134 \mathrm{Ps}, 133 \mathrm{Qs}, 133 \mathrm{Rs}$

ANSWER :

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

ANSWER :

(A) 1

PROBLEM 18 :

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5 pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located 2023 pads to the right of her starting position?
(A) 405
(B) 407
(C) 409
(D) 411
(E) 413

ANSWER :

(D) 411

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) $4: 9$

ANSWER :

PROBLEM 20 :

Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
(A) 56
(B) 57
(C) 58
(D) 60
(E) 61

ANSWER :

(D) 60

PROBLEM 21 :

Alina writes the numbers $1,2, \ldots, 9$ on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

PROBLEM 22 :

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is 4000 . What is the first term?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 10

ANSWER :

(D) 5

PROBLEM 23 :

Each square in a $3 \times 3$ grid is randomly filled with one of the 4 gray and white tiles shown below on the right.

What is the probability that the tiling will contain a large gray diamond in one of the smaller $2 \times 2$ grids? Below is an example of such tiling.

(A) $\frac{1}{1024}$
(B) $\frac{1}{256}$
(C) $\frac{1}{64}$
(D) $\frac{1}{16}$
(E) $\frac{1}{4}$

ANSWER :

PROBLEM 24 :

Isosceles $\triangle A B C$ has equal side lengths $A B$ and $B C$. In the figure 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 of $h$ of $\triangle A B C$ ? (Diagram not drawn to scale.)

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

ANSWER :

(A) 14.6

PROBLEM 25 :

Fifteen integers $a_1, a_2, a_3, \ldots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that

$$
1 \leq a_1 \leq 10,13 \leq a_2 \leq 20, \text { and } 241 \leq a_{15} \leq 250 .
$$

What is the sum of digits of $a_{14}$ ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

SOLUTION :

(A) 8

AMERICAN MATHEMATICS COMPETITION 8 - 2024

PROBLEM 1 :

What is the unit digit of:

$$
222,222-22,222-2,222-222-22-2 ?
$$

(A) 0
(B) 2
(C) 4
(D) 8
(E) 10

ANSWER :

(B) 2

PROBLEM 2 :

What is the value of this expression in decimal form?

$$
\frac{44}{11}+\frac{110}{44}+\frac{44}{1100}
$$

(A) 6.4
(B) 6.504
(C) 6.54
(D) 6.9
(E) 6.94

ANSWER :

PROBLEM 3 :

Four squares of side length $4,7,9$, and 10 are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?

(A) 42
(B) 45
(C) 49
(D) 50
(E) 52

ANSWER :

(E) 52

PROBLEM 4 :

When Yunji added all the integers from 1 to 9 , she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER :

(E) 9

PROBLEM 5 :

Aaliyah rolls two standard 6 -sided dice. She notices that the product of the two numbers rolled is a multiple of 6 . Which of the following integers cannot be the sum of the two numbers?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER :

(B) 6

PROBLEM 6 :

Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P, Q, R$, and $S$. What is the sorted order of the four paths from shortest to longest?

(A) $P, Q, R, S$
(B) $P, R, S, Q$
(C) $Q, S, P, R$
(D) $R, P, S, Q$
(E) $R, S, P, Q$

ANSWER :

(D) $R, P, S, Q$

PROBLEM 7 :

A $3 \times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2 \times 2,1 \times 4$, and $1 \times 1$, shown below. What is the minimum possible number of $1 \times 1$ tiles used?

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

ANSWER :

(E) 5

PROBLEM 8 :

On Monday, Taye has $\$ 2$. Every day, he either gains $\$ 3$ or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

ANSWER :

(D) 6

PROBLEM 9 :

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles, and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

ANSWER :

(E) 28

PROBLEM 10 :

In January 1980 the Mauna Loa Observatory recorded carbon dioxide (CO2) levels of 338 ppm (parts per million). Over the years the average $C O 2$ reading has increased by about 1.515 ppm each year. What is the expected $C O 2$ level in ppm in January 2030 ? Round your answer to the nearest integer.
(A) 399
(B) 414
(C) 420
(D) 444
(E) 459

ANSWER :

(B) 414

PROBLEM 11 :

The coordinates of $\triangle A B C$ are $A(5,7), B(11,7)$, and $C(3, y)$, with $y>7$. The area of $\triangle A B C$ is 12 . What is the value of $y$ ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

ANSWER :

(D) 11

PROBLEM 12 :

Rohan keeps 90 guppies in 4 fish tanks.

There is 1 more guppy in the 2 nd tank than in the 1 st tank.
There are 2 more guppies in the 3rd tank than in the 2nd tank.
There are 3 more guppies in the 4th tank than in the 3rd tank.

How many guppies are in the 4th tank?
(A) 20
(B) 21
(C) 23
(D) 24
(E) 26

ANSWER :

(E) 26

PROBLEM 13 :

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz Bunny start on the ground, make a sequence of 6 hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

(A) 4
(B) 5
(C) 6
(D) 8
(E) 12

ANSWER :

(B) 5

PROBLEM 14 :

The one-way routes connecting towns $A, M, C, X, Y$, and $Z$ are shown in the figure below(not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?

(A) 28
(B) 29
(C) 30
(D) 31
(E) 32

ANSWER :

(A) 28

PROBLEM 15 :

Let the letters $F, L, Y, B, U, G$ represent distinct digits. Suppose $\underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}$ is the greatest number that satisfies the equation

$$
8 \cdot \underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}=\underline{B} \underline{U} \underline{G} \underline{B} \underline{U} \underline{G} .
$$

What is the value of $\underline{F} \underline{L} \underline{Y}+\underline{B} \underline{U} \underline{G}$ ?
(A) 1089
(B) 1098
(C) 1107
(D) 1116
(E) 1125

ANSWER :

PROBLEM 16 :

Minh enters the numbers 1 through 81 into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3 ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

ANSWER :

(D) 11

PROBLEM 17 :

A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a 3 $x 3$ grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?

(A) 20
(B) 24
(C) 27
(D) 28
(E) 32

ANSWER :

(E) 32

PROBLEM 18 :

Three concentric circles centered at $O$ have radii of 1, 2, and 3 . Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angles $B O C$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle B O C$ in degrees?

(A) 108
(B) 120
(C) 135
(D) 144
(E) 150

ANSWER :

(A) 108

PROBLEM 19 :

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?

(A) 0
(B) $\frac{1}{5}$
(C) $\frac{4}{15}$
(D) $\frac{1}{3}$
(E) $\frac{2}{5}$

ANSWER :

PROBLEM 20 :

Any three vertices of the cube $P Q R S T U V W$, shown in the figure below, can be connected to form a triangle. (For example, vertices $P, Q$, and $R$ can be connected to form isosceles $\triangle P Q R$.) How many of these triangles are equilateral and contain $P$ as a vertex?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 6

ANSWER :

(D) 3

PROBLEM 21 :

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3: 1$. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is $4: 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
(A) 10
(B) 12
(C) 16
(D) 20
(E) 24

ANSWER :

(E) 24

PROBLEM 22 :

A roll of tape is 4 inches in diameter and is wrapped around a ring that is 2 inches in diameter. A cross section of the tape is shown in the figure below. The tape is 0.015 inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 100 inches.

(A) 300
(B) 600
(C) 1200
(D) 1500
(E) 1800

ANSWER :

(B) 600

PROBLEM 23 :

Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the 4 cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time?

(A) 6000
(B) 6500
(C) 7000
(D) 7500
(E) 8000

ANSWER :

PROBLEM 24 :

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is 8 feet high while the other peak is 12 feet high. Each peak forms a $90^{\circ}$ angle, and the straight sides form a $45^{\circ}$ angle with the ground. The artwork has an area of 183 square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$ ?

(A) 4
(B) 5
(C) $4 \sqrt{2}$
(D) 6
(E) $5 \sqrt{2}$

ANSWER :

(B) 5

PROBLEM 25 :

A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?

(A) $\frac{8}{15}$
(B) $\frac{32}{55}$
(C) $\frac{20}{33}$
(D) $\frac{34}{55}$
(E) $\frac{8}{11}$

ANSWER :