AMERICAN MATHEMATICS COMPETITION 10 A - 2018

Problem 1

What is the value of

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

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

Answer:

(B) $\frac{11}{7}$

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 Alica 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.

Answer:

(A) Liliane has $20 \%$ more soda than Alice.

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) Febuary 11
(E) Febuary 12

Answer:

(E) Febuary 12

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

Answer:

(E) 24

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 $d$ ?
(A) $(0,4)$
(B) $(4,5)$
(C) $(4,6)$
(D) $(5,6)$
(E) $(5, \infty)$

Answer:

(D) $(5,6)$

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

Answer:

(B) 300

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

Answer:

(E) 9

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

Answer:

Problem 9

All of the triangles in the diagram below are similar to iscoceles triangle $A B C$, in which $A B=A C$. Each of the 7 smallest triangles has area 1, and $\triangle A B C$ has area 40 . What is the area of trapezoid $D B C E$ ?

(A) 16
(B) 18
(C) 20
(D) 22
(E) 24

Answer:

(E) 24

Problem 10

Suppose that real number $x$ satisfies

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

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

Answer:

(A) 8

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 $n$ ?
(A) 42
(B) 49
(C) 56
(D) 63
(E) 84

Answer:

(E) 84

Problem 12

How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations?

$$
\begin{array}{r}
x+3 y=3 \
||x|-|y||=1
\end{array}
$$

(A) 1
(B) 2
(C) 3
(D) 4
(E) 8

Answer:

Problem 13

A paper triangle with sides of lengths 3,4 , and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?

(A) $1+\frac{1}{2} \sqrt{2}$
(B) $\sqrt{3}$
(C) $\frac{7}{4}$
(D) $\frac{15}{8}$
(E) 2

Answer:

(D) $\frac{15}{8}$

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

Answer:

(A) 80

Problem 15

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $A B$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

(A) 21
(B) 29
(C) 58
(D) 69
(E) 93

Answer:

(D) 69

Problem 16

Right triangle $A B C$ has leg lengths $A B=20$ and $B C=21$. Including $\overline{A B}$ and $\overline{B C}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{A C}$ ?
(A) 5
(B) 8
(C) 12
(D) 13
(E) 15

Answer:

(D) 13

Problem 17

Let $S$ be a set of 6 integers taken from ${1,2, \ldots, 12}$ 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 values of an element in $S$ ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 7

Answer:

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

Answer:

(D) 3281

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

Answer:

(E) $\frac{2}{5}$

Problem 20

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of $90^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
(A) 510
(B) 1022
(C) 8190
(D) 8192
(E) 65,534

Answer:

(B) 1022

Problem 21

Which of the following describes the set of values of $a$ for which the curves $x^{2}+y^{2}=a^{2}$ and $y=x^{2}-a$ in the real $x y$-plane intersect at exactly 3 points?
(A) $a=\frac{1}{4}$
(B) $\frac{1}{4}\frac{1}{4}$
(D) $a=\frac{1}{2}$
(E) $a>\frac{1}{2}$

Answer:

(E) $a>\frac{1}{2}$

Problem 22

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

Answer:

(D) 13

Problem 23

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the fiels is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted?

(A) $\frac{25}{27}$
(B) $\frac{26}{27}$
(C) $\frac{73}{75}$
(D) $\frac{145}{147}$
(E) $\frac{74}{75}$

Answer:

(D) $\frac{145}{147}$

Problem 24

Triangle $A B C$ with $A B=50$ and $A C=10$ has area 120 . Let $D$ be the midpoint of $\overline{A B}$, and let $E$ be the midpoint of $\overline{A C}$. The angle bisector of $\angle B A C$ intersects $\overline{D E}$ and $\overline{B C}$ at $F$ and $G$, respectively. What is the area of quadrilateral $F D B G$ ?
(A) 60
(B) 65
(C) 70
(D) 75
(E) 80

Answer:

(D) 75

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) 14
(C) 16
(D) 18
(E) 20

Answer:

(D) 18

AMERICAN MATHEMATICS COMPETITION 10 A - 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

Answer:

Problem 2

Pablo buys popsicles for his friends. The store sells single popsicles for $\$ 1$ each, 3popsicle 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

Answer:

(D) 13

Problem 3

Tamara has three rows of two 6 -feet by 2 -feet flower beds in her garden. The beds are separated and also surrounded by 1 -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?

(A) 72
(B) 78
(C) 90
(D) 120
(E) 150

Answer:

(B) 78

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

Answer:

(B) 14

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

Answer:

Problem 6

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of of these statements necessarily follows logically?
(A) If Lewis did not receive an A , then he got all of the multiple choice questions wrong.
(B) If Lewis did not receive an A , then he got at least one of the multiple choice questions wrong.
(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A .
(D) If Lewis received an A , then he got all of the multiple choice questions right.
(E) If Lewis received an A , then he got at least one of the multiple choice questions right.

Answer:

(B) If Lewis did not receive an A , then he got at least one of the multiple choice questions wrong.

Problem 7

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
(A) $30 \%$
(B) $40 \%$
(C) $50 \%$
(D) $60 \%$
(E) $70 \%$

Answer:

(A) $30 \%$

Problem 8

At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
(A) 240
(B) 245
(C) 290
(D) 480
(E) 490

Answer:

(B) 245

Problem 9

Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph , and uphill at 5 kph . Penny rides on a flat road at 30 kph , downhill at 40 kph , and uphill at 10 kph . Minnie goes from town A to town B, a distance of 10 km all uphill, then from town B to town C, a distance of 15 km all downhill, and then back to town A, a distance of 20 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 $45-\mathrm{km}$ ride than it takes Penny?
(A) 45
(B) 60
(C) 65
(D) 90
(E) 95

Answer:

Problem 10

Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm . She places the rods with lengths $3 \mathrm{~cm}, 7 \mathrm{~cm}$, and 15 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

Answer:

(B) 17

Problem 11

The region consisting of all point in three-dimensional space within 3 units of line segment $A B$ has volume $216 \pi$. What is the length $A B$ ?
(A) 6
(B) 12
(C) 18
(D) 20
(E) 24

Answer:

(D) 20

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

Answer:

(E) three rays with a common endpoint

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 $0,1,1,2,0,2, \cdots$ 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

Answer:

(D) 9

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) $25 \%$\[0pt]

Answer:

(D) $23 \%$

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

Answer:

Problem 16

There are 10 horses, named Horse 1, Horse 2, . . . 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

Answer:

(B) 3

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) $5 \sqrt{2}$

Answer:

(D) 7

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 $q-p$ ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

(D) 4

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?
(A)12
(B)16
(C) 28
(D) 32
(E) 40

Answer:

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

Answer:

(D) 1239

Problem 21

A square with side length $x$ is inscribed in a right triangle with sides of length 3,4 , and 5 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length 3,4 , and 5 so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$ ?
(A) $\frac{12}{13}$
(B) $\frac{35}{37}$
(C) 1
(D) $\frac{37}{35}$
(E) $\frac{13}{12}$

Answer:

(D) $\frac{37}{35}$

Problem 22

Sides $\overline{A B}$ and $\overline{A C}$ of triangle $A B C$ are tangent to a circle as points $B$ and $C$, respectively. What fraction of the area of $\triangle A B C$ lies outside the circle?
(A) $\frac{4 \sqrt{3} \pi}{27}-\frac{1}{3}$
(B) $\frac{\sqrt{3}}{2}-\frac{\pi}{8}$
(C) $\frac{1}{2}$
(D) $\sqrt{3}-\frac{2 \sqrt{3} \pi}{9}$
(E) $\frac{4}{3}-\frac{4 \sqrt{3} \pi}{27}$

Answer:

(E) $\frac{4}{3}-\frac{4 \sqrt{3} \pi}{27}$

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

Answer:

(B) 2148

Problem 24

For certain real numbers $a, b$, and $c$, the polynomial $g(x)=x^{3}+a x^{2}+x+10$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial
\end{enumerate}

$$
f(x)=x^{4}+x^{3}+b x^{2}+100 x+c
$$

What is $f(1)$ ?
(A) -9009
(B) -8008
(C) -7007
(D) -6006
(E) -5005

Answer:

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

Answer:

(A) 226

American Mathematics Competition 10A - 2021

Problem 1
What is the value of $\frac{(2112-2021)^{2}}{169}$ ?
(A) 7
(B) 21
(C) 49
(D) 64
(E) 91

Answer:

Problem 2
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by 1 inch, the card would have area 18 square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by 1 inch?
(A) 16
(B) 17
(C) 18
(D) 19
(E) 20

Answer:

(E) 20

Problem 3
What is the maximum number of balls of clay with radius 2 that can completely fit inside a cube of side length 6 assuming that the balls can be reshaped but not compressed before they are packed in the cube?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer:

(D) 6

Problem 4
Mr. Lopez has a choice of two routes to get to work. Route A is 6 miles long, and his average speed along this route is 30 miles per hour. Route B is 5 miles long, and his average speed along this route is 40 miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is 20 miles per hour. By how many minutes is Route B quicker than Route A?
(A) $2 \frac{3}{4}$
(B) $3 \frac{3}{4}$
(C) $4 \frac{1}{2}$
(D) $5 \frac{1}{2}$
(E) $6 \frac{3}{4}$

Answer:

(B) $3 \frac{3}{4}$

Problem 5
The six-digit number $\underline{2} \underline{2} \underline{1} \underline{0} \underline{\mathrm{~A}}$ is prime for only one digit A . What is A ?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Answer:

(E) 9

Problem 6
Elmer the emu takes 44 equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in 12 equal leaps. The telephone poles are evenly spaced, and the 41st pole along this road is exactly one mile ( 5280 feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
(A) 6
(B) 8
(C) 10
(D) 11
(E) 15

Answer:

(B) 8

Problem 7
As shown in the figure below, point $E$ lies in the opposite half-plane determined by line $C D$ from point $A$ so that $\angle C D E=110^{\circ}$. Point $F$ lies on $\overline{A D}$ so that $D E=D F$, and $A B C D$ is a square. What is the degree measure of $\angle A F E$ ?

(A) 160
(B) 164
(C) 166
(D) 170
(E) 174

Answer:

(D) 170

Problem 8
A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

(B) 1

Problem 9
When a certain unfair die is rolled, an even number is 3 times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?
(A) $\frac{3}{8}$
(B) $\frac{4}{9}$
(C) $\frac{5}{9}$
(D) $\frac{9}{16}$
(E) $\frac{5}{8}$

Answer:

(E) $\frac{5}{8}$

Problem 10
A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student is picked at random and the number of students in their class, including that student, is noted. What is $t-s$ ?
(A) -18.5
(B) -13.5
(C) 0
(D) 13.5
(E) 18.5

Answer:

(B) -13.5

Problem 11
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts 210 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 42 steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
(A) 70
(B) 84
(C) 98
(D) 105
(E) 126

Answer:

(A) 70

Problem 12
The base-nine representation of the number $N$ is $27,006,000,052_{\text {nine }}$. What is the remainder when $N$ is divided by 5 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

(D) 3

Problem 13
Each of 6 balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other 5 balls?
(A) $\frac{1}{64}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{5}{16}$
(E) $\frac{1}{2}$

Answer:

(D) $\frac{5}{16}$

Problem 14
How many ordered pairs $(x, y)$ of real numbers satisfy the following system of equations?

(A) 1
(B) 2
(C) 3
(D) 5
(E) 7

Answer:

(D) 5

Problem 15

Isosceles triangle $A B C$ has $A B=A C=3 \sqrt{6}$, and a circle with radius $5 \sqrt{2}$ is tangent to line $A B$ at $B$ and to line $A C$ at $C$. What is the area of the circle that passes through vertices $A, B$, and $C$ ?

(A) $24 \pi$
(B) $25 \pi$
(C) $26 \pi$
(D) $27 \pi$
(E) $28 \pi$

Answer:

Problem 16

The graph of $f(x)=|\lfloor x\rfloor|-|\lfloor 1-x\rfloor|$ is symmetric about which of the following?
(A) the $y$-axis
(B) the line $x=1$
(C) the origin
(D) the point $\left(\frac{1}{2}, 0\right)$
(E) the point $(1,0)$

Answer:

(D) the point $\left(\frac{1}{2}, 0\right)$

Problem 17

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $A B C D E F$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A, B$, and $C$ are 12, 9, and 10 meters, respectively. What is the height, in meters, of the pillar at $E$ ?

(A) 9
(B) $6 \sqrt{3}$
(C) $8 \sqrt{3}$
(D) 17
(E) $12 \sqrt{3}$

Answer:

(D) 17

Problem 18
A farmer's rectangular field is partitioned into a 2 by 2 grid of 4 rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?

(A) 12
(B) 64
(C) 84
(D) 90
(E) 144

Answer:

Problem 19
A disk of radius 1 rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius 1 rolls all the way around the outside of the same square and sweeps out a region of area $2 A$. The value of $s$ can be written as $a+\frac{b \pi}{c}$, where $a, b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$ ?

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

Answer:

(A) 10

Problem 20

For how many ordered pairs ( $b, c$ ) of positive integers does neither $x^{2}+ b x+c=0$ nor $x^{2}+c x+b=0$ have two distinct real solutions?

(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

Answer:

(B) 6

Problem 21

Each of 20 balls is tossed independently and at random into one of 5 bins. Let $p$ be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Let $q$ be the probability that every bin ends up with 4 balls. What is $\frac{p}{q}$ ?
(A) 1
(B) 4
(C) 8
(D) 12
(E) 16

Answer:

(E) 16

Problem 22

Inside a right circular cone with base radius 5 and height 12 are three congruent spheres each with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$ ?

(A) $\frac{3}{2}$
(B) $\frac{90-40 \sqrt{3}}{11}$
(C) 2
(D) $\frac{144-25 \sqrt{3}}{44}$
(E) $\frac{5}{2}$

Answer:

(B) $\frac{90-40 \sqrt{3}}{11}$

Problem 23

For each positive integer $n$, let $f_{1}(n)$ be twice the number of positive integer divisors of $n$, and for $j \geq 2$, let $f_{j}(n)=f_{1}\left(f_{j-1}(n)\right)$. For how many values of $n \leq 50$ is $f_{50}(n)=12$ ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Answer:

(D) 10

Problem 24

Each of the 12 edges of a cube is labeled 0 or 1 . Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the 6 faces of the cube equal to 2 ?
(A) 8
(B) 10
(C) 12
(D) 16
(E) 20

Answer:

(E) 20

Problem 25

A quadratic polynomial $p(x)$ with real coefficients and leading coefficient 1 is called disrespectful if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$ ?
(A) $\frac{5}{16}$
(B) $\frac{1}{2}$
(C) $\frac{5}{8}$
(D) 1
(E) $\frac{9}{8}$

Answer:

(A) $\frac{5}{16}$

American Mathematics Competition 10A - 2025

Problem 1

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1: 30$, traveling due north at a steady 8 miles per hour. Betsy leaves on her bicycle from the same point at $2: 30$, traveling due east at a steady 12 miles per hour. At what time will they be exactly the same distance from their common starting point?
(A) $3: 30$
(B) $3: 45$
(C) $4: 00$
(D) $4: 15$
(E) $4: 30$

Answer:

(E) $4: 30$

Problem 2

A box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second nut mix containing 20 percent peanuts, 40 percent cashews, and 40 percent almonds is added to the box resulting in a new nut mix that is 40 percent peanuts. How many pounds of cashews are now in the box?
(A) 3.5
(B) 4
(C) 4.5
(D) 5
(E) 6

Answer:

(B) 4

Problem 3

How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025 ?
(A) 2025
(B) 2026
(C) 3012
(D) 3037
(E) 4050

Answer:

(D) 3037

Problem 4

A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is 15 . Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from 12 to 14 . If Ash plays with the teachers, the average age on that team will decrease from 55 to 52 . How old is Ash?
(A) 28
(B) 29
(C) 30
(D) 32
(E) 33

Answer:

(A) 28

Problem 5

Consider the sequence of positive integers

$$
1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2 \ldots
$$

What is the 2025th term in the sequence?
(A) 5
(B) 15
(C) 16
(D) 44
(E) 45

Answer:

(E) 45

Problem 6

In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle $20^{\circ}$-angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
(A) 80
(B) 90
(C) 100
(D) 110
(E) 120

Answer:

Problem 7

Suppose $a$ and $b$ are real numbers. When the polynomial $x^{3}+x^{2}+a x+b$ is divided by $x-1$, the remainder is 4 . When the polynomial is divided by $x-2$, the remainder is 6 . What is $b-a$ ?
(A) 14
(B) 15
(C) 16
(D) 17
(E) 18

Answer:

(E) 18

Problem 8

Agnes writes the following four statements on a blank piece of paper.

Each statement is either true or false. How many false statements did Agnes write on the paper?\
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

(B) 1

Problem 9

Let $f(x)=100 x^{3}-300 x^{2}+200 x$. For how many real numbers $a$ does the graph of $y=f(x-a)$ pass through the point $(1,25)$ ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) more than 4

Answer:

Problem 10

A semicircle has diameter $A B$ and chord $C D$ of length 16 parallel to $A B$. A smaller circle with diameter on $A B$ and tangent to $C D$ is cut from the larger semicircle, as shown below.

What is the area of the resulting figure, shown shaded?
(A) $16 \pi$
(B) $24 \pi$
(C) $32 \pi$
(D) $48 \pi$
(E) $64 \pi$

Answer:

Problem 11

The sequence $1, x, y, z$ is arithmetic. The sequence $1, p, q, z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x+y+z+p+q$ ?
(A) 66
(B) 91
(C) 103
(D) 132
(E) 149

Answer:

(E) 149

Problem 12

Carlos uses a 4-digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is 0 . How many 4 -digit passcodes satisfy these conditions?
(A) 176
(B) 192
(C) 432
(D) 464
(E) 608

Answer:

(D) 464

Problem 13

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k$, where $0<k<1$. The spaces between squares are alternately shaded, as shown in the figure (which is not necessarily drawn to scale).

The area of the shaded portion of the figure is $64 \%$ of the area of the original square. What is $k$ ?
(A) $\frac{3}{5}$
(B) $\frac{16}{25}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
(E) $\frac{4}{5}$

Answer:

(D) $\frac{3}{4}$

Problem 14

Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
(A) $\frac{1}{6}$
(B) $\frac{1}{5}$
(C) $\frac{2}{9}$
(D) $\frac{3}{13}$
(E) $\frac{1}{4}$

Answer:

(B) $\frac{1}{5}$

Problem 15

In the figure below, $A B E F$ is a rectangle, $\quad \overline{A D} \perp \overline{D E} \quad, \quad A F=7 \quad, \quad A B=1 \quad$, and $\quad A D=5 \quad$. What is the area of $\triangle A B C$ ?

(A) $\frac{3}{8}$
(B) $\frac{4}{9}$
(C) $\frac{1}{8} \sqrt{13}$
(D) $\frac{7}{15}$
(E) $\frac{1}{8} \sqrt{15}$

Answer:

(A) $\frac{3}{8}$

Problem 16

There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?
(A) $\frac{4}{3}$
(B) $\frac{13}{9}$
(C) $\frac{5}{3}$
(D) $\frac{17}{9}$
(E) 2

Answer:

(D) $\frac{17}{9}$

Problem 17

Let $N$ be the unique positive integer such that dividing 273436 by $N$ leaves a remainder of 16 and dividing 272760 by $N$ leaves a remainder of 15 . What is the tens digit of $N$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

(E) 4

Problem 18

The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of $4,4,5$ is

$$
\frac{1}{\frac{1}{3}\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{5}\right)}=\frac{30}{7} .
$$

What is the harmonic mean of all the real roots of the $4050{ }^{\text {th }}$ degree polynomial

$$
\prod_{k=1}^{2025}\left(k x^{2}-4 x-3\right)=\left(x^{2}-4 x-3\right)\left(2 x^{2}-4 x-3\right)\left(3 x^{2}-4 x-3\right) \cdots\left(2025 x^{2}-4 x-3\right) ?
$$

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

Answer:

(B) $-\frac{3}{2}$

Problem 19

An array of numbers is constructed beginning with the numbers $-1 \quad 3 \quad 1$ in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with -1 and 1 , respectively.

If the process continues, one of the rows will sum to 12,288 . In that row, what is the third number from the left?
(A) -29
(B) -21
(C) -14
(D) -8
(E) -3

Answer:

(A) -29

Problem 20

A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and $g>0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of $g$ can\
be written as $\frac{a \sqrt{b}-c}{d}$, where $a, b, c$, and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$. What is $a+b+c+d$ ?
(A) 119
(B) 120
(C) 121
(D) 122
(E)123

Answer:

(A) 119

Problem 21

A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$, is not an element of the set. For example, ${1,4,6}$ and the empty set are sum-free, but ${2,4,5}$ is not. What is the greatest possible number of elements in a sum-free subset of ${1,2,3, \ldots, 20}$.
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Answer:

Problem 22

A circle of radius $r$ is surrounded by three circles, whose radii are 1,2 , and 3 , all externally tangent to the inner circle and to each other, as shown.

What is $r$ ?
(A) $\frac{1}{4}$
(B) $\frac{6}{23}$
(C) $\frac{3}{11}$
(D) $\frac{5}{17}$
(E) $\frac{3}{10}$

Answer:

(B) $\frac{6}{23}$

Problem 23
Triangle $\triangle A B C$ has side lengths $A B=80, B C=45$, and $A C=75$. The bisector $\angle B$ and the altitude to side $\overline{A B}$ intersect at point $P$. What is $B P$ ?
(A) 18
(B) 19
(C) 20
(D) 21
(E) 22

Answer:

(D) 21

Problem 24

Call a positive integer fair if no digit is used more than once, it has no 0 s , and no digit is adjacent to two greater digits. For example, 196,23 and 12463 are fair, but 1546,320 , and 34321 are not. How many fair positive integers are there?
(A) 511
(B) 2584
(C) 9841
(D) 17711
(E) 19682

Answer:

Problem 25

A point $P$ is chosen at random inside square $A B C D$. the probability that $\overline{A P}$ is neither the shortest nor the longest side of $\triangle A P B$ can be written

$$
\frac{a+b \pi-c \sqrt{d}}{e}
$$

, where $a, b, c, d, \quad$ and $\quad e \quad$ are positive integers, $\operatorname{gcd}(a, b, c, e)=1$, and $d$ is not divisible by the square of a prime. What is $a+b+c+d+e$ ?
(A) 25
(B) 26
(C) 27
(D) 28
(E) 29

Answer:

(A) 25

AMERICAN MATHEMATICS COMPETITION 8 - 2010

Problem 1

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 in Mr. Newton, and 9 in Mrs. Young's class are taking the AMC 8 this year. How many mathematics students at Euclid High School are taking the contest?
(A) 26
(B) 27
(C) 28
(D) 29
(E) 30

Answer:

Problem 2

If $a @ b=\frac{a \times b}{a+b}$, for $a, b$ positive integers, then what is $5 @ 10$ ?
(A) $\frac{3}{10}$
(B) 1
(C) 2
(D) $\frac{10}{3}$
(E) 50

Answer:

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

Problem 3

3 The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?

(A) 50
(B) 62
(C) 70
(D) 89
(E) 100

Answer:

Problem 4

What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$ ?
(A) 6.5
(B) 7
(C) 7.5
(D) 8.5
(E) 9

Answer:

Problem 5

Alice needs to replace a light bulb located 10 centimeters below the ceiling of her kitchen. The ceiling is 2.4 meters above the floor. Alice is 1.5 meters tall and can reach 46 centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
(A) 32
(B) 34
(C) 36
(D) 38
(E) 40

Answer:

(B) 34

Problem 6

Which of the following has the greatest number of line of symmetry?
(A) Equilateral Triangle (B) Non-square rhombus (C) Non-square rectangle (D) Isosceles Triangle (E) Square

Answer:

(E) Square

Problem 7

Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar?
(A) 6
(B) 10
(C) 15
(D) 25
(E) 99

Answer:

(B) 10

Problem 8

As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1 / 2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1 / 2$ mile behind her. Emily rides at a constant rate of 12 miles per hour. Ermenson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Ermenson?
(A) 6
(B) 8
(C) 12
(D) 15
(E) 16

Answer:

(D) 15

Problem 9

Ryan got $80 \%$ of the problems on a 25 -problem test, $90 \%$ on a 40 -problem test, and $70 \%$ on a 10 -problem test. What percent of all problems did Ryan answer correctly?
(A) 64
(B) 75
(C) 80
(D) 84
(E) 86

Answer:

(D) 84

Problem 10

6 pepperoni circles will exactly fit across the diameter of a 12 -inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{5}{6}$
(E) $\frac{7}{8}$

Answer:

(B) $\frac{2}{3}$

Problem 11

The top of one tree is 16 feet higher than the top of another tree. The height of the 2 trees are at a ratio of $3: 4$. In feet, how tall is the taller tree?
(A) 48
(B) 64
(C) 80
(D) 96
(E) 112

Answer:

(B) 64

Problem 12

12 & Of the 500 balls in a large bag, $80 \%$ are red and the rest are blue. How many of the red balls must be removed so that $75 \%$ of the remaining balls are red?
(A) 25
(B) 50
(C) 75
(D) 100
(E) 150

Answer:

(D) 100

Problem 13

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30 \%$ of the perimeter. What is the length of the longest side?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Answer:

(E) 11

Problem 14

What is the sum of the prime factors of 2010 ?
(A) 67
(B) 75
(C) 77
(D) 201
(E) 210

Answer:

Problem 15

A jar contains 5 different colors of gumdrops. $30 \%$ are blue, $20 \%$ are brown, $15 \%$ red, $10 \%$ yellow, and the other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
(A) 35
(B) 36
(C) 42
(D) 48
(E) 64

Answer:

Problem 16

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
(B) $\sqrt{\pi}$
(C) $\pi$
(D) $2 \pi$
(E) $\pi^{2}$

Answer:

(B) $\sqrt{\pi}$

Problem 17

The diagram shows an octagon consisting of 10 unit squares. The portion below $\overline{P Q}$ is a unit square and a triangle with base 5 . If $\overline{P Q}$ bisects the area of the octagon, what is the ratio $\frac{X Q}{Q Y}$ ?

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

Answer:

(D) $\frac{2}{3}$

Problem 18

A decorative window is made up of a rectangle with semicircles at either end. The ratio of $A D$ to $A B$ is $3: 2$. And $A B$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

(A) $2: 3$
(B) $3: 2$
(C) $6: \pi$
(D) $9: \pi$
(E) $30: \pi$

Answer:

Problem 19

The two circles pictured have the same center $C$. Chord $\overline{A D}$ is tangent to the inner circle at $B, A C$ is 10 , and chord $\overline{A D}$ has length 16 . What is the area between the two circles?

(A) $36 \pi$
(B) $49 \pi$
(C) $64 \pi$
(D) $81 \pi$
(E) $100 \pi$

Answer:

Problem 20

In a room, $2 / 5$ of the people are wearing gloves, and $3 / 4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
(A) 3
(B) 5
(C) 8
(D) 15
(E) 20

Answer:

(A) 3

Problem 21

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, she read $1 / 5$ of the pages plus 12 more, and on the second day she read $1 / 4$ of the remaining pages plus 15 more. On the third day she read $1 / 3$ of the remaining pages plus 18 more. She then realizes she has 62 pages left, which she finishes the next day. How many pages are in this book?
(A) 120
(B) 180
(C) 240
(D) 300
(E) 360

Answer:

Problem 22

The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8

Answer:

(E) 8

Problem 23

Semicircles $P O Q$ and $R O S$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$ ?

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

Answer:

(B) $\frac{1}{2}$

Problem 24

What is the correct ordering of the three numbers, $10^{8}, 5^{12}$, and $2^{24}$ ?
(A) $2^{24}<10^{8}<5^{12}$
(B) $2^{24}<5^{12}<10^{8}$
(C) $5^{12}<2^{24}<10^{8}$ (D) $10^{8}< 5^{12}<2^{24}$ (E) $10^{8}<2^{24}<5^{12}$

Answer:

(A) $2^{24}<10^{8}<5^{12}$

Problem 25

Everyday at school, Jo climbs a flight of 6 stairs. Joe can take the stairs 1,2, or 3 at a time. For example, Jo could climb 3, then 1 , then 2 . In how many ways can Jo climb the stairs?
(A) 13
(B) 18
(C) 20
(D) 22
(E) 24

Answer:

(E) 24

American Mathematics Competition - 2006

Problem 1

Mindy made three purchases for $\$ 1.98, \$ 5.04$ and $\$ 9.89$. What was her total, to the nearest dollar?
(A) $\$ 10$
(B) $\$ 15$
(C) $\$ 16$
(D) $\$ 17$
(E) $\$ 18$

Answer:

(D) $\$ 17$

Problem 2

On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5 . What is his score?
(A) 1
(B) 6
(C) 13
(D) 19
(E) 26

Answer:

Problem 3

Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?
(A) $\frac{1}{2}$
(B) $\frac{3}{4}$
(C) 1
(D) 2
(E) 3

Answer:

(A) $\frac{1}{2}$

Problem 4

Initially, a spinner points west. Chenille moves it clockwise $2 \frac{1}{4}$ revolutions and then counterclockwise $3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?

(A) north
(B) east
(C) south
(D) west
(E) northwest

Answer:

(B) east

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

Answer:

(D) 30

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

Answer:

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) $X, Z, Y$

Answer:

(B) $Z, X, Y$

Problem 8

The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station?

(A) 39
(B) 48
(C) 52
(D) 55
(E) 75

Answer:

(E) 75

Problem 9

What is the product of $\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2006}{2005}$ ?
(A) 1
(B) 1002
(C) 1003
(D) 2005
(E) 2006

Answer:

Problem 10

Jorge's teacher asks him to plot all the ordered pairs ( $w, l$ ) of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12 . What should his graph look like?

Answer:

Problem 11

How many two-digit numbers have digits whose sum is a perfect square?
(A) 13
(B) 16
(C) 17
(D) 18
(E) 19

Answer:

Problem 12

Antonette gets $70 \%$ on a 10 -problem test, $80 \%$ on a 20 -problem test and $90 \%$ on a 30 -problem test. If the three tests are combined into one 60 -problem test, which percent is closest to her overall score?
(A) 40
(B) 77
(C) 80
(D) 83
(E) 87

Answer:

(D) 83

Problem 13

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62 -mile route between Escanaba and Marquette. At what time in the morning do they meet?
(A) $10: 00$
(B) $10: 15$
(C) $10: 30$
(D) $11: 00$
(E) $11: 30$

Answer:

(D) $11: 00$

Problem 14

Problems 14, 15 and 16 involve Mrs. Reed's English assignment.

{A Novel Assignment}
The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?

(A) 7,600
(B) 11,400
(C) 12,500
(D) 15,200
(E) 22,800

Answer:

(B) 11,400

Problem 15

The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?
(A) 425
(B) 444
(C) 456
(D) 484
(E) 506

Answer:

Problem 16

Before Chandra and Bob start reading, Alice says she would like to team read
league Education Center
with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
(A) 6400
(B) 6600
(C) 6800
(D) 7000
(E) 7200

Answer:

(B) 6600

Problem 17

Jeff rotates spinners $P, Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number?

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

Answer:

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

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) $\frac{19}{27}$

Answer:

(D) $\frac{5}{9}$

Problem 19

Triangle $A B C$ is an isosceles triangle with $\overline{A B}=\overline{B C}$. Point $D$ is the midpoint of both $\overline{B C}$ and $\overline{A E}$, and $\overline{C E}$ is 11 units long. Triangle $A B D$ is congruent to triangle $E C D$. What is the length of $\overline{B D}$ ?

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

Answer:

(D) 5.5

Problem 20

A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer:

Problem 21

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm . The aquarium is tilled with water to a depth of 37 cm . A rock with volume $1000 \mathrm{~cm}^{3}$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
(A) 0.25
(B) 0.5
(C) 1
(D) 1.25
(E) 2.5

Answer:

(A) 0.25

Problem 22

Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

Answer:

(D) 26

Problem 23

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

Answer:

(A) 0

Problem 24

In the multiplication problem below, $A, B, C$ and $D$ are different digits. What A B A\
is $A+B$ ?

Answer:

(A) 1

Problem 25

Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?

(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

Answer:

(B) 14

American Mathematics Competition - 2012

Problem 1

Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic?

Answer:

(E) 9.

Problem 2

In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?

(A) 600
(B) 700
(C) 800
(D) 900
(E) 1000

Answer:

(B) 700.

Problem 3

On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise as 6:57 am, and the sunset as 8:15 pm. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
(A) 5:10 PM
(B) 5:21 PM
(C) 5:41 PM
(D) 5: 57 PM
(E) 6:03 PM

Answer:

(B) 5:21 PM.

Problem 4

Peter's family ordered a 12 -slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?

(A) $\frac{1}{24}$
(B) $\frac{1}{12}$
(C) $\frac{1}{8}$
(D) $\frac{1}{6}$
(E) $\frac{1}{4}$

Answer:

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 X, in centimeters?

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

Answer:

(E) 5.

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

Answer:

(E) 88.

Problem 7

Isabella must take four 100 -point tests in her math class. Her goal is to achieve an average grade of at least 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized that she could still reach her goal. What is the lowest possible score she could have made on the third test?

(A) 90
(B) 92
(C) 95
(D) 96
(E) 97

Answer:

(B) 92.

Problem 8

A shop advertises that everything is "half price in today's sale." In addition, a coupon gives a $20 \%$ discount on sale prices. Using the coupon, the price today represents what percentage discount off the original price?

(A) 10
(B) 33
(C) 40
(D) 60
(E) 70

Answer:

(D) 60.

Problem 9

The Fort Worth Zoo has a number of two-legged birds and a number of fourlegged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?

(A) 61
(B) 122
(C) 139
(D) 150
(E) 161

Answer:

Problem 10

How many 4 -digit numbers greater than 1000 are there that use the four digits of 2012?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 12

Answer:

(D) 9.

Problem 11

The mean, median, and unique mode of the positive integers $3,4,5,6,6,7, x$ are all equal. What is the value of $x$ ?
(A) 5
(B) 6
(C) 7
(D) 11
(E) 12

Answer:

(D) 11.

Problem 12

What is the units digit of $13^{2012}$ ?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Answer:

(A) 1.

Problem 13

Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$ 1.43$. Sharona bought some of the same pencils and paid $\$ 1.87$. How many more pencils did Sharona buy than Jamar?

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

Answer:

Problem 14

In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Answer:

(B) 7.

Problem 15

The smallest number greater than 2 that leaves a remainder of 2 when divided by $3,4,5$, or 6 lies between what numbers?

(A) 40 and 50
(B) 51 and 55
(C) 56 and 60
(D) 61 and 65
(E) 66 and 99

Answer:

(D) 61 and 65.

Problem 16

Each of the digits $0,1,2,3,4,5,6,7,8$, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?

(A) 76531
(B) 86724
(C) 87431
(D) 96240
(E) 97403

Answer:

Problem 17

A square with an 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

Answer:

(B) 4.

Problem 18

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50 ?

(A) 3127
(B) 3133
(C) 3137
(D) 3139
(E) 3149

Answer:

(A) 3127.

Problem 19

In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?

(A) 6
(B) 8
(C) 9
(D) 10
(E) 18

Answer:

Problem 20

What is the correct ordering of the three numbers $\frac{5}{19}, \frac{7}{21}$, and $\frac{9}{23}$, in increasing order?

(A) $\frac{9}{23}<\frac{7}{21}<\frac{5}{19}$
(B) $\frac{5}{19}<\frac{7}{21}<\frac{9}{23}$
(C) $\frac{9}{23}<\frac{5}{19}<\frac{7}{21}$
(D) $\frac{5}{19}<\frac{9}{23}<\frac{7}{21}$
(E) $\frac{7}{21}<\frac{5}{19}<\frac{9}{23}$

Answer:

(B) $\frac{5}{19}<\frac{7}{21}<\frac{9}{23}$

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) $5 \sqrt{2}$
(B) 10
(C) $10 \sqrt{2}$
(D) 50
(E) $50 \sqrt{2}$

Answer:

(D) 50.

Problem 22

Let $R$ be a set of nine distinct integers. Six of the elements of the set are 2, 3, 4, 6,9 , and 14 . What is the number of possible values of the median of $R$ ?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Answer:

(D) 7.

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) $4 \sqrt{3}$
(E) $6 \sqrt{3}$

Answer:

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) $\frac{\sqrt{2}}{\pi}$
(D) $\frac{\pi-1}{\pi}$
(E) $\frac{3}{\pi}$

Answer:

(A) $\frac{4-\pi}{\pi}$

Problem 25

A square with area 4 is inscribed in a square with area 5 , with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$ and the other of length $b$. What is the value of $a b$ ?

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

Answer:

IOQM 2025 Questions, Answer Key, Solutions

Answer Key

Answer 1 40	Answer 2 17	Answer 3 18	Answer 4 5	Answer 5 36
Answer 6 18	Answer 7 576	Answer 8 44	Answer 9 28	Answer 10 15
Answer 11 80	Answer 12 38	Answer 13 13	Answer 14 11	Answer 15 75
Answer 16 8	Answer 17 8	Answer 18 1	Answer 19 72	Answer 20 42
Answer 21 80	Answer 22 7	Answer 23 19	Answer 24 66	Answer 25 9
Answer 26 6	Answer 27 37	Answer 28 12	Answer 29 33	Answer 30 97

Problem 1

If $60 \%$ of a number $x$ is 40 , then what is $x \%$ of 60 ?

Problem 2

Find the number of positive integers $n$ less than or equal to 100 , which are divisible by 3 but are not divisible by 2.

Problem 3

The area of an integer-sided rectangle is 20 . What is the minimum possible value of its perimeter?

Problem 4

How many isosceles integer-sided triangles are there with perimeter 23?

Problem 5

How many 3 -digit numbers $a b c$ in base 10 are there with $a \neq 0$ and $c=a+b$ ?

Problem 6

The height and the base radius of a closed right circular cylinder are positive integers and its total surface area is numerically equal to its volume. If its volume is $k \pi$ where $k$ is a positive integer, what is the smallest possible value of $k$ ?

Problem 7

A quadrilateral has four vertices $A, B, C, D$. We want to colour each vertex in one of the four colours red, blue, green or yellow, so that every side of the quadrilateral and the diagonal $A C$ have end points of different colours. In how many ways can we do this?

Problem 8

The sum of two real numbers is a positive integer $n$ and the sum of their squares is $n+1012$. Find the maximum possible value of $n$.

Problem 9

Four sides and a diagonal of a quadrilateral are of lengths $10, 20, 28, 50, 75$, not necessarily in that order. Which amongst them is the only possible length of the diagonal?

Problem 10

The age of a person (in years) in 2025 is a perfect square. His age (in years) was also a perfect square in 2012. His age (in years) will be a perfect cube $m$ years after 2025. Determine the smallest value of $m .=15$

Problem 11

There are six coupons numbered 1 to 6 and six envelopes, also numbered 1 to 6 . The first two coupons are placed together in any one envelope. Similarly, the third and the fourth are placed together in a different envelope, and the last two are placed together in yet another different envelope. How many ways can this be done if no coupon is placed in the envelope having the same number as the coupon?

Problem 12

Consider five-digit positive integers of the form $\overline{a b c a b}$ that are divisible by the two digit number $a b$ but not divisible by 13 . What is the largest possible sum of the digits of such a number?

Problem 13

A function $f$ is defined on the set of integers such that for any two integers $m$ and $n$,

$$
f(m n+1)=f(m) f(n)-f(n)-m+2
$$

holds and $f(0)=1$. Determine the largest positive integer $N$ such that $\sum_{k=1}^N f(k)<100$ .

Problem 14

Consider a fraction $\frac{a}{b} \neq \frac{3}{4}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$ and $b \leq 15$. If this fraction is chosen closest to $\frac{3}{4}$ amongst all such fractions, then what is the value of $a+b$ ?

Problem 15

Three sides of a quadrilateral are $a=4 \sqrt{3}, b=9$ and $c=\sqrt{3}$. The sides $a$ and $b$ enclose an angle of $30^{\circ}$, and the sides $b$ and $c$ enclose an angle of $90^{\circ}$. If the acute angle between the diagonals is $x^{\circ}$, what is the value of $x$ ?

Problem 16

$f(x)$ and $g(x)$ be two polynomials of degree 2 such that

$$
\frac{f(-2)}{g(-2)}=\frac{f(3)}{g(3)}=4
$$

If $g(5)=2, f(7)=12, g(7)=-6$, what is the value of $f(5)$ ?

Problem 17

The triangle $A B C, \angle B=90^{\circ}, A B=1$ and $B C=2$. On the side $B C$ there are two points $D$ and $E$ such that $E$ lies between $C$ and $D$ and $D E F G$ is a square, where $F$ lies on $A C$ and $G$ lies on the circle through $B$ with centre $A$. If the area of $D E F G$ is $\frac{m}{n}$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 18

$M T A I$ is a parallelogram of area $\frac{40}{41}$ square units such that $M I=1 / M T$. If $d$ is the least possible length of the diagonal $M A$, and $d^2=\frac{a}{b}$, where $a, b$ are positive integers with $\operatorname{gcd}(a, b)=1$, find $|a-b|$.

Problem 19

Let $N$ be the number of nine-digit integers that can be obtained by permuting the digits of 223334444 and which have at least one 3 to the right of the right-most occurrence of 4 . What is the remainder when $N$ is divided by $100$?

Problem 20

Let $f$ be the function defined by

$$
f(n)=\text { remainder when } n^n \text { is divided by } 7,
$$

for all positive integers $n$. Find the smallest positive integer $T$ such that $f(n+T)=f(n)$ for all positive integers $n$.

Problem 21

Let $P(x)=x^{2025}, Q(x)=x^4+x^3+2 x^2+x+1$. Let $R(x)$ be the polynomial remainder when the polynomial $P(x)$ is divided by the polynomial $Q(x)$. Find $R(3)$.

Problem 22

Let $A B C D$ be a rectangle and let $M, N$ be points lying on sides $A B$ and $B C$, respectively. Assume that $M C= C D$ and $M D=M N$, and that points $C, D, M, N$ lic on a circle. If $(A B / B C)^2=m / n$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 23

Let $A B C D$ be a rectangle and let $M, N$ be points lying on sides $A B$ and $B C$, respectively. Assume that $M C= C D$ and $M D=M N$, and that points $C, D, M, N$ lie on a circle. If $(A B / B C)^2=m / n$ where $m$ and $n$ are positive integers with $\operatorname{gcd}(m, n)=1$, what is the value of $m+n$ ?

Problem 24

There are $m$ blue marbles and $n$ red marbles on a table. Armaan and Babita play a game by taking turns. In each turn the player has to pick a marble of the colour of his/her choice. Armaan starts first, and the player who picks the last red marble wins. For how many choices of $(m, n)$ with $1 \leq m, n \leq 11$ can Armaan force a win?

Problem 25

For some real numbers $m, n$ and a positive integer $a$, the list $(a+1) n^2, m^2, a(n+1)^2$ consists of three consecutive integers written in increasing order. What is the largest possible value of $m^2$ ?

Problem 26

Let $S$ be a circle of radius 10 with centre $O$. Suppose $S_1$ and $S_2$ are two circles which touch $S$ internally and intersect each other at two distinct points $A$ and $B$. If $\angle O A B=90^{\circ}$ what is the sum of the radii of $S_1$ and $S_2$ ?

Solution

Problem 27

A regular polygon with $n \geq 5$ vertices is said to be colourful if it is possible to colour the vertices using at most 6 colours such that each vertex is coloured with exactly one colour, and such that any 5 consecutive vertices have different colours. Find the largest number $n$ for which a regular polygon with $n$ vertices is not colourful.

Solution

Problem 28

Find the number of ordered triples $(a, b, c)$ of positive integers such that $1 \leq a, b, c \leq 50$ which satisfy the relation

$$
\frac{\operatorname{lcm}(a, c)+\operatorname{lcm}(b, c)}{a+b}=\frac{26 c}{27}
$$

Here, by $\operatorname{lcm}(x, y)$ we mean the LCM, that is, least common multiple of $x$ and $y$.

Problem 29

Consider a sequence of real numbers of finite length. Consecutive four term averages of this sequence are strictly increasing, but consecutive seven term averages are strictly decreasing. What is the maximum possible length of such a sequence?

Problem 30

Assume $a$ is a positive integer which is not a perfect square. Let $x, y$ be non-negative integers such that $\sqrt{x-\sqrt{x+a}}=\sqrt{a}-y$. What is the largest possible value of $a$ such that $a<100 ?$

BHASKARA Contest - NMTC - Screening Test – 2025

Problem 1

The greatest 4 -digit number such that when divided by 16,24 and 36 leaves 4 as remainder in each case is
А) 9994
B) 9940
C) 9094
D) 9904

Problem 2

$A B C D$ is a rectangle whose length $A B$ is 20 units and breadth is 10 units. Also, given $A P=8$ units. The area of the shaded region is $\frac{p}{q}$ sq unit, where $p, q$ are natural numbers with no common factors other than 1 . The value of $p+q$ is
A) 167
В) 147
C) 157
D) 137

Problem 3

The solution of $\frac{\sqrt[7]{12+x}}{x}+\frac{\sqrt[7]{12+x}}{12}=\frac{64}{3}(\sqrt[7]{x})$ is of the form $\frac{a}{b}$ where $a, b$ are natural numbers with $\operatorname{GCD}(a, b)=1$; then $(b-a)$ is equal to
A) 115
B) 114
C) 113
D) 125

Problem 4

The value of $(52+6 \sqrt{43})^{3 / 2}-(52-6 \sqrt{43})^{3 / 2}$ is
A) 858
В) 918
C) 758
D) 828

Problem 5

In the adjoining figure $\angle D C E=10^{\circ}$, $\angle C E D=98^{\circ}, \angle B D F=28^{\circ}$
Then the measure of angle $x$ is
A) $72^{\circ}$
B) $76^{\circ}$
C) $44^{\circ}$
D) $82^{\circ}$

Problem 6

$A B C$ is a right triangle in which $\angle \mathrm{B}=90^{\circ}$. The inradius of the triangle is $r$ and the circumradius of the triangle is R . If $\mathrm{R}: r=5: 2$, then the value of $\cot ^2 \frac{A}{2}+\cot ^2 \frac{C}{2}$ is
A) $\frac{25}{4}$
B) 17
C) 13
D) 14

Problem 7

If $(\alpha, \beta)$ and $(\gamma, \beta)$ are the roots of the simultaneous equations:

\[
|x-1|+|y-5|=1 ; \quad y=5+|x-1|
\]

then the value of $\alpha+\beta+\gamma$ is
A) $\frac{15}{2}$
B) $\frac{17}{2}$
C) $\frac{14}{3}$
D) $\frac{19}{2}$

Problem 8

Three persons Ram, Ali and Peter were to be hired to paint a house. Ram and Ali can paint the whole house in 30 days, Ali and Peter in 40 days while Peter and Ram can do it in 60 days. If all of them were hired together, in how many days can they all three complete $50 \%$ of the work?
A) $24 \frac{1}{3}$
B) $25 \frac{1}{2}$
C) $26 \frac{1}{3}$
D) $26 \frac{2}{3}$

Problem 9

$\frac{\sqrt{a+3 b}+\sqrt{a-3 b}}{\sqrt{a+3 b}-\sqrt{a-3 b}}=x$, then the value of $\frac{3 b x^2+3 b}{a x}$ is
A) 1
B) 2
C) 3
D) 4

Problem 10

The number of integral solutions of the inequation $\left|\frac{2}{x-13}\right|>\frac{8}{9}$ is
A) 1
B) 2
C) 3
D) 4

Problem 11

In the adjoining figure, $P$ is the centre of the first circle, which touches the other circle in C . PCD is along the diameter of the second circle. $\angle \mathrm{PBA}=20^{\circ}$ and $\angle \mathrm{PCA}=30^{\circ}$.

The tangents at B and D meet at E . The measure of the angle $x$ is
A) $75^{\circ}$
B) $80^{\circ}$
C) $70^{\circ}$
D) $85^{\circ}$

Problem 12

If $\alpha, \beta$ are the values of $x$ satisfying the equation $3 \sqrt{\log _2 x}-\log _2 8 x+1=0$, where $\alpha<\beta$, then the value of $\left(\frac{\beta}{\alpha}\right)$ is
A) 2
B) 4
C) 6
D) 8

Problem 13

When a natural number is divided by 11 , the remainder is 4 . When the square of this number is divided by 11 , the remainder is
A) 4
B) 5
C) 7
D) 9

Problem 14

The unit's digit of a 2-digit number is twice the ten's digit. When the number is multiplied by the sum of the digits the result is 144 . For another 2-digit number, the ten's digit is twice the unit's digit and the product of the number with the sum of its digits is 567 . Then the sum of the two 2 -digit numbers is
A) 68
В) 86
C) 98
D) 87

Problem 15

$A B C D E$ is a pentagon. $\angle A E D=126^{\circ}, \angle B A E=\angle C D E$ and $\angle A B C$ is $4^{\circ}$ less than $\angle B A E$ and $\angle B C D$ is $6^{\circ}$ less than $\angle C D E . P R, Q R$ the bisectors of $\angle B P C, \angle E Q D$ respectively, meet at $R$. Points $\mathrm{P}, \mathrm{C}, \mathrm{D}, \mathrm{Q}$ are collinear. Then measure of $\angle P R Q$ is
A) $151^{\circ}$
B) $137^{\circ}$
C) $141^{\circ}$
D) $143^{\circ}$

Problem 16

$a, b, c$ are real numbers such that $b-c=8$ and $b c+a^2+16=0$.
The numerical value of $a^{2025}+b^{2025}+c^{2025}$ is $\rule{2cm}{0.2mm}$.

Problem 17

Given $f(x)=\frac{2025 x}{x+1}$ where $x \neq-1$. Then the value of $x$ for which $f(f(x))=(2025)^2$ is $\rule{2cm}{0.2mm}$.

Problem 18

The sum of all the roots of the equation $\sqrt[3]{16-x^3}=4-x$ is $\rule{2cm}{0.2mm}$.

Problem 19

In the adjoining figure, two
Quadrants are touching at $B$.
$C E$ is joined by a straight line, whose mid-point is $F$.

The measure of $\angle C E D$ is $\rule{2cm}{0.2mm}$.

Problem 20

The value of $k$ for which the equation $x^3-6 x^2+11 x+(6-k)=0$ has exactly three positive integer solutions is $\rule{2cm}{0.2mm}$.

Problem 21

The number of 3-digit numbers of the form $a b 5$ (where $a, b$ are digits) which are divisible by 9 is $\rule{2cm}{0.2mm}$.

Problem 22

If $a=\sqrt{(2025)^3-(2023)^3}$, the value of $\sqrt{\frac{a^2-2}{6}}$ is $\rule{2cm}{0.2mm}$.

Problem 23

In a math Olympiad examination, $12 \%$ of the students who appeared from a class did not solve any problem; $32 \%$ solved with some mistakes. The remaining 14 students solved the paper fully and correctly. The number of students in the class is $\rule{2cm}{0.2mm}$.

Problem 24

When $a=2025$, the numerical value of
$\left|2 a^3-3 a^2-2 a+1\right|-\left|2 a^3-3 a^2-3 a-2025\right|$ is $\rule{2cm}{0.2mm}$.

Problem 25

A circular hoop and a rectangular frame are standing on the level ground as shown. The diagonal $A B$ is extended to meet the circular hoop at the highest point $C$. If $A B=18 \mathrm{~cm}, B C=32 \mathrm{~cm}$, the radius of the hoop (in cm ) is $\rule{2cm}{0.2mm}$.

Problem 26

' $n$ ' is a natural number. The number of ' $n$ ' for which $\frac{16\left(n^2-n-1\right)^2}{2 n-1}$ is a natural number is $\rule{2cm}{0.2mm}$.

Problem 27

The number of solutions $(x, y)$ of the simultaneous equations $\log _4 x-\log _2 y=0, \quad x^2=8+2 y^2$ is $\rule{2cm}{0.2mm}$.

Problem 28

In the adjoining figure,
$P A, P B$ are tangents.
$A R$ is parallel to $P B$

$P Q=6 ; Q R=18 .$

Length $S B= \rule{2cm}{0.2mm}$.

Problem 29

A large watermelon weighs 20 kg with $98 \%$ of its weight being water. It is left outside in the sunshine for some time. Some water evaporated and the water content in the watermelon is now $95 \%$ of its weight in water. The reduced weight in kg is $\rule{2cm}{0.2mm}$.

Problem 30

In a geometric progression, the fourth term exceeds the third term by 24 and the sum of the second and third term is 6 . Then, the sum of the second, third and fourth terms is $\rule{2cm}{0.2mm}$.

Australian Mathematics Competition - 2019 - Upper Primary Division - Grades 5 & 6- Questions and Solutions

Problem 1:

$201-9=$

(A) 111 (B) 182 (C) 188 (D) 192 (E) 198

Problem 2:

A Runnyball team has 5 players. This graph shows the number of goals each player scored in a tournament. Who scored the second-highest number of goals?

(A) Ali (B) Beth (C) Caz (D) Dan (E) Evan

Problem 3:

Six million two hundred and three thousand and six would be written as

(A) 62036 (B) 6230006 (C) 6203006 (D) 6203600 (E) 6200306

Problem 4:

These cards were dropped on the table, one at a time. In which order were they dropped?

Problem 5:

Sophia is at the corner of 1st Street and 1st Avenue. Her school is at the corner of 4th Street and 3rd Avenue. To get there, she walks

(A) 4 blocks east, 3 blocks north (B) 3 blocks west, 4 blocks north (C) 4 blocks west, 2 blocks north (D) 3 blocks east, 2 blocks north (E) 2 blocks north, 2 blocks south

Problem 6:

Jake is playing a card game, and these are his cards. Elena chooses one card from Jake at random. Which of the following is Elena most likely to choose?

Problem 7:

Which 3D shape below has 5 faces and 9 edges?

Problem 8:

We're driving from Elizabeth to Renmark, and as we leave we see this sign. We want to stop at a town for lunch and a break, approximately halfway to Renmark. Which town is the best place to stop?

(A) Gawler (B) Nuriootpa (C) Truro (D) Blanchetown (E) Waikerie

Problem 9:

What is the difference between the heights of the two flagpoles, in metres?

(A) 16.25 (B) 16.75 (C) 17.25 (D) 17.75 (E) 33.25

Problem 10:

Most of the numbers on this scale are missing.

Which number should be at position $P$ ?
(A) 18 (B) 33 (C) 34 (D) 36 (E) 42

Problem 11:

In a game, two ten-sided dice each marked 0 to 9 are rolled and the two uppermost numbers are added. For example, with the dice as shown, $0+9=9$. How many different results can be obtained?

(A) 17 (B) 18 (C) 19 (D) 20 (E) 21

Problem 12:

Every row and every column of this $3 \times 3$ square must contain each of the numbers 1,2 and 3 . What is the value of $N+M$ ?

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

Problem 13:

Ada Lovelace and Charles Babbage were pioneering researchers into early mechanical computers. They were born 24 years apart.

To the nearest year, how much longer did Charles Babbage live than Ada Lovelace?

(A) 29 (B) 32 (C) 35 (D) 37 (E) 43

Problem 14:

You have 12 metres of ribbon. Each decoration needs $\frac{2}{5}$ of a metre of ribbon. How many decorations can you make?

(A) 6 (B) 7 (C) 10 (D) 24 (E) 30

Problem 15:

Andrew and Bernadette are clearing leaves from their backyard. Bernadette can rake the backyard in 60 minutes, while Andrew can do it in 30 minutes with the vacuum setting on the leaf blower. If they work together, how many minutes will it take?

(A) 10 (B) 20 (C) 24 (D) 30 (E) 45

Problem 16:

A carpet tile measures 50 cm by 50 cm . How many of these tiles would be needed to cover the floor of a room 6 m long and 4 m wide?

(A) 24 (B) 20 (C) 40 (D) 48 (E) 96

Problem 17:

In how many different ways can you place the numbers 1 to 4 in these four circles so that no two consecutive numbers are side by side?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 12

Problem 18:

John, Chris, Anne, Holly and Mike are seated around a round table, each with a card with a number on it in front of them. Each person can see the numbers in front of their two neighbours. Each person calls out the sum of the two numbers in front of their neighbours. John says 30, Chris says 33, Anne says 31, Holly says 38 and Mike says 36. Holly has the number 21 in front of her. What number does Anne have in front of her?

(A) 9 (B) 13 (C) 15 (D) 18 (E) 19

Problem 19:

Annabel has 2 identical equilateral triangles. Each has an area of $9 \mathrm{~cm}^2$. She places one triangle on top of the other as shown to form a star, as shown. What is the area of the star in square centimetres?

(A) 10 (B) 12 (C) 14 (D) 16 (E) 18

Problem 20:

Lola went on a train trip. During her journey she slept for $\frac{3}{4}$ of an hour and stayed awake for $\frac{3}{4}$ of the journey. How long did the trip take?

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

Problem 21:

My sister and I are playing a game where she picks two counting numbers and I have to guess them. When I tell her a number, she multiplies my number by her first number and then adds her second number. When I say 15 , she says 50 . When I say 2 , she says 11 . If I say 6 , what should she say?

(A) 23 (B) 27 (C) 35 (D) 41 (E) 61

Problem 22:

Once the muddy water from the 2018 Ingham floods had drained from Harry's house, he found this folded map that had been standing in the floodwater at an angle. He unfolded it and laid it out to dry, but it was still mud-stained. What could it look like now?

Problem 23:

A tower is built from exactly 2019 equal rods. Starting with 3 rods as a triangular base, more rods are added to form a regular octahedron with this base as one of its faces. The top face is then the base of the next octahedron. The diagram shows the construction of the first three octahedra. How many octahedra are in the tower when it is finished?

(A) 2016 (B) 1008 (C) 336 (D) 224 (E) 168

Problem 24:

These three cubes are labelled in exactly the same way, with the 6 letters A, M, C, D, E and F on their 6 faces:

The cubes are now placed in a row so that the front looks like this:

When we look at the cubes from the opposite side, we will see

Problem 25:

In Jeremy's hometown of Windar, people live in either North, East, South, West or Central Windar. Jeremy is putting together a chart showing where the students in his class live, but unfortunately his dog chewed his survey results before he managed to label the five columns.

He only remembers two things about the survey: South Windar is more common than both East and Central Windar, and the number of students in North and Central Windar combined is the same as the total of the other three regions.
Using only this information, how many columns can Jeremy correctly label with $100 \%$ certainty?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 5

Problem 26:

Pip starts with a large square sheet of paper and makes two straight cuts to form four smaller squares. She then takes one of these smaller squares and makes two more straight cuts to make four even smaller ones, as shown.

Continuing in this way, how many cuts does Pip need to make to get a total of 1000 squares of various sizes?

Problem 27:

Seven of the numbers from 1 to 9 are placed in the circles in the diagram in such a way that the products of the numbers in each vertical or horizontal line are the same. What is this product?

Problem 28:

A hare and a tortoise compete in a 10 km race. The hare runs at $30 \mathrm{~km} / \mathrm{h}$ and the tortoise walks at $3 \mathrm{~km} / \mathrm{h}$. Unfortunately, at the start, the hare started running in the opposite direction. After some time, it realised its mistake and turned round, catching the tortoise at the halfway mark. For how many minutes did the hare run in the wrong direction?

Problem 29:

I want to place the numbers 1 to 10 in this diagram, with one number in each circle. On each of the three sides, the four numbers add to a side total, and the three side totals are all the same. What is the smallest number that this side total could be?

Problem 30:

The sum of two numbers is 11.63 . When adding the numbers together, Oliver accidentally shifted the decimal point in one of the numbers one position to the left. Oliver got an answer of 5.87 instead. What is one hundred times the difference between the two original numbers?