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

(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 - 2020

Problem 1

What value of $\boldsymbol{x}$ satisfies

$$
x-\frac{3}{4}=\frac{5}{12}-\frac{1}{3} ?
$$

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

Answer:

(E) $\frac{5}{6}$

Problem 2
The numbers $3,5,7, a$ and $b$ have an average (arithmetic mean) of 15 . What is the average of $a$ and $b$ ?
(A) 0
(B) 15
(C) 30
(D) 45
(E) 60

Answer:

Problem 3
Assuming $a \neq 3, b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression?

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

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

Answer:

(A) -1

Problem 4
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $\$ 0.50$ per mile, and her only expense is gasoline at $\$ 2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
(A) 20
(B) 22
(C) 24
(D) 25
(E) 26

Answer:

(E) 26

Problem 5
What is the sum of all real numbers $\boldsymbol{x}$ for which

$$
\left|x^{2}-12 x+34\right|=2 ?
$$

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

Answer:

Problem 6
How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5 ?
(A) 80
(B) 100
(C) 125
(D) 200
(E) 500

Answer:

(B) 100

Problem 7
The 25 integers from -10 to 14 inclusive, can be arranged to form a 5 -by- 5 square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
(A) 2
(B) 5
(C) 10
(D) 25
(E) 50

Answer:

Problem 8
What is the value of

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

(A) 9,800
(B) 9,900
(C) 10,000
(D) 10,100
(E) 10,200

Answer:

(B) 9,900

Problem 9
A single bench section at a school event can hold either 7 adults or 11 children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N$ ?
(A) 9
(B) 18
(C) 27
(D) 36
(E) 77

Answer:

(B) 18

Problem 10
Seven cubes, whose volumes are $1,8,27,64,125,216$, and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
(A) 644
(B) 658
(C) 664
(D) 720
(E) 749

Answer:

(B) 658

Problem 11
What is the median of the following list of 4040 numbers?

$$
1,2,3, \ldots, 2020,1^{2}, 2^{2}, 3^{2}, \ldots, 2020^{2}
$$

(A) 1974.5
(B) 1975.5
(C) 1976.5
(D) 1977.5
(E) 1978.5

Answer:

Problem 12
Triangle $A M C$ is isosceles with $A M=A C$. Medians $\overline{M V}$ and $\overline{C U}$ are perpendicular to each other, and $M V=C U=12$. What is the area of $\triangle A M C$ ?

(A) 48
(B) 72
(C) 96
(D) 144
(E) 192

Answer:

Problem 13
A frog sitting at the point $(1,2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1 , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0),(0,4),(4,4)$, and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?
(A) $\frac{1}{2}$
(B) $\frac{5}{8}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
(E) $\frac{7}{8}$

Answer:

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

Problem 14
Real numbers $\boldsymbol{x}$ and $\boldsymbol{y}$ satisfy

$$
x+y=4 \text { and } x \cdot y=-2 .
$$

What is the value of

$$
x+\frac{x^{3}}{y^{2}}+\frac{y^{3}}{x^{2}}+y ?
$$

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

Answer:

(D) 440

Problem 15
A positive integer divisor of 12 ! is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{\boldsymbol{m}}{\boldsymbol{n}}$, where $m$ and $n$ are relatively prime positive integers. What is $\boldsymbol{m}+\boldsymbol{n}$ ?
(A) 3
(B) 5
(C) 12
(D) 18
(E) 23

Answer:

(E) 23

Problem 16
A point is chosen at random within the square in the coordinate plane whose vertices are ( 0,0 ), $(2020,0),(2020,2020)$, and $(0,2020)$. The probability that the point is within $\boldsymbol{d}$ units of a lattice point is $\frac{\mathbf{1}}{\mathbf{2}}$. (A point $(\boldsymbol{x}, \boldsymbol{y})$ is a lattice point if $\boldsymbol{x}$ and $\boldsymbol{y}$ are both integers.) What is $\boldsymbol{d}$ to the nearest tenth?
(A) 0.3
(B) 0.4
(C) 0.5
(D) 0.6
(E) 0.7

Answer:

(B) 0.4

Problem 17
Define

$$
P(x)=\left(x-1^{2}\right)\left(x-2^{2}\right) \cdots\left(x-100^{2}\right) .
$$

How many integers $\boldsymbol{n}$ are there such that

$$
P(n) \leq 0 ?
$$

(A) 4900
(B) 4950
(C) 5000
(D) 5050
(E) 5100

Answer:

(E) 5100

Problem 18
Let ( $a, b, c, d$ ) be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}$. For how many such quadruples is it true that $a \cdot d-b \cdot c$ is odd? (For example, ( $0,3,1,1$ ) is one such quadruple, because $0 \cdot 1-3 \cdot 1=-3$ is odd.)
(A) 48
(B) 64
(C) 96
(D) 128
(E) 192

Answer:

Problem 19
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

(A) 125
(B) 250
(C) 405
(D) 640
(E) 810

Answer:

(E) 810

Problem 20
Quadrilateral $A B C D$ satisfies

$$
\angle A B C=\angle A C D=90^{\circ}, A C=20, \text { and } C D=30 .
$$

Diagonals $\overline{A C}$ and $\overline{B D}$ intersect at point $E$, and $A E=5$. What is the area of quadrilateral $A B C D$ ?
(A) 330
(B) 340
(C) 350
(D) 360
(E) 370

Answer:

(D) 360

Problem 21
There exists a unique strictly increasing sequence of nonnegative integers

$$
a_{1}<a_{2}<\ldots<a_{k}
$$

such that

$$
\frac{2^{289}+1}{2^{17}+1}=2^{a_{1}}+2^{a_{2}}+\ldots+2^{a_{k}}
$$

\section*{American Mathematics Competitions}
What is $\boldsymbol{k}$ ?
(A) 117
(B) 136
(C) 137
(D) 273
(E) 306

Answer:

Problem 22
For how many positive integers $n \leq 1000$ is

$$
\left\lfloor\frac{998}{n}\right\rfloor+\left\lfloor\frac{999}{n}\right\rfloor+\left\lfloor\frac{1000}{n}\right\rfloor
$$

not divisible by 3 ? (Recall that $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)
(A) 22
(B) 23
(C) 24
(D) 25
(E) 26

Answer:

(A) 22

Problem 23
Let $T$ be the triangle in the coordinate plane with vertices $(0,0),(4,0)$, and $(0,3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the 125 sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
(A) 12
(B) 15
(C) 17
(D) 20
(E) 25

Answer:

(A) 12

Problem 24
Let $\boldsymbol{n}$ be the least positive integer greater than 1000 for which

$$
\operatorname{gcd}(63, n+120)=21 \quad \text { and } \quad \operatorname{gcd}(n+63,120)=60 .
$$

What is the sum of the digits of $\boldsymbol{n}$ ?
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

Answer:

Problem 25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7 . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
(A) $\frac{7}{36}$
(B) $\frac{5}{24}$
(C) $\frac{2}{9}$
(D) $\frac{17}{72}$
(E) $\frac{1}{4}$

Answer:

(A) $\frac{7}{36}$

American Mathematics Competition 10A - 2019

Problem 1

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

Answer:

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

Answer:

(A) 0

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

Answer:

(D) 12

Problem 4
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?
(A) 75
(B) 76
(C) 79
(D) 84
(E) 91

Answer:

(B) 76

Problem 5
What is the greatest number of consecutive integers whose sum is 45 ?
(A) 9
(B) 25
(C) 45
(D) 90
(E) 120

Answer:

(D) 90

Problem 6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

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

Answer:

Problem 7
Two lines with slopes $\frac{1}{2}$ and 2 intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$ ?
(A) 4
(B) $4 \sqrt{2}$
(C) 6
(D) 8
(E) $6 \sqrt{2}$

Answer:

Problem 8
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

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

Answer:

Problem 9
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers?
(A) 995
(B) 996
(C) 997
(D) 998
(E) 999

Answer:

(B) 996

Problem 10
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
(A) 17
(B) 25
(C) 26
(D) 27
(E) 28

Answer:

Problem 11
How many positive integer divisors of $201^{9}$ are perfect squares or perfect cubes (or both)?
(A) 32
(B) 36
(C) 37
(D) 39
(E) 41

Answer:

Problem 12
Melanie computes the mean $\mu$, the median $M$, and the modes of the 365 values that are the dates in the months of 2019 . Thus her data consist of $121 \mathrm{~s}, 122 \mathrm{~s}$, . . . $, 1228 \mathrm{~s}, 1129 \mathrm{~s}, 1130 \mathrm{~s}$, and 731 s . Let $d$ be the median of the modes. Which of the following statements is true?
(A) $\mu<d<M$
(B) $M<d<\mu$
(C) $d=M=\mu$
(D) $d<M<\mu$
(E) $d<\mu<M$

Answer:

(E) $d<\mu<M$

Problem 13
Let $\triangle A B C$ be an isosceles triangle with $B C=A C$ and $\angle A C B=40^{\circ}$. Contruct the circle with diameter $\overline{B C}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{A C}$ and $\overline{A B}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $B C D E$. What is the degree measure of $\angle B F C$ ?
(A) 90
(B) 100
(C) 105
(D) 110
(E) 120

Answer:

(D) 110

Problem 14
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$ ?
(A) 14
(B) 16
(C) 18
(D) 19
(E) 21

Answer:

(D) 19

Problem 15
A sequence of numbers is defined recursively by $a_{1}=1, a_{2}=\frac{3}{7}$, and

for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q$ ?
(A) 2020
(B) 4039
(C) 6057
(D) 6061
(E) 8078

Answer:

(E) 8078

Problem 16
The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius 1 ?

(A) $4 \pi \sqrt{3}$
(B) $7 \pi$
(C) $\pi(3 \sqrt{3}+2)$
(D) $10 \pi(\sqrt{3}-1)$
(E) $\pi(\sqrt{3}+6)$

Answer:

(A) $4 \pi \sqrt{3}$

Problem 17
A child builds towers using identically shaped cubes of different color. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.)
(A) 24
(B) 288
(C) 312
(D) 1, 260
(E) 40,320

Answer:

(D) 1, 260

Problem 18
For some positive integer $k$, the repeating base- $k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0 . \overline{23}_{k}=0.232323 \ldots k$. What is $k$ ?
(A) 13
(B) 14
(C) 15
(D) 16
(E) 17

Answer:

(D) 16

Problem 19
What is the least possible value of

$$
(x+1)(x+2)(x+3)(x+4)+2019
$$

where $x$ is a real number?
(A) 2017
(B) 2018
(C) 2019
(D) 2020
(E) 2021

Answer:

(B) 2018

Problem 20
The numbers $1,2, \ldots, 9$ are randomly placed into the 9 squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
(A) $1 / 21$
(B) $1 / 14$
(C) $5 / 63$
(D) $2 / 21$
(E) $1 / 7$

Answer:

(B) $1 / 14$

Problem 21
A sphere with center $O$ has radius 6 . A triangle with sides of length 15, 15, and 24 is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
(A) $2 \sqrt{3}$
(B) 4
(C) $3 \sqrt{2}$
(D) $2 \sqrt{5}$
(E) 5

Answer:

(D) $2 \sqrt{5}$

Problem 22
Real numbers between 0 and 1 , inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that

$$
|x-y|>\frac{1}{2} ?
$$

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

Answer:

(B) $\frac{7}{16}$

Problem 23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number 1, then Todd must say the next two numbers ( 2 and 3 ), then Tucker must say the next three numbers $(4,5,6)$, then Tadd must say the next four numbers $(7,8,9,10)$, and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number 10,000 is reached. What is the 2019th number said by Tadd?
(A) 5743
(B) 5885
(C) 5979
(D) 6001
(E) 6011

Answer:

Problem 24
Let $p, q$, and $r$ be the distinct roots of the polynomial $x^{3}-22 x^{2}+80 x-67$. It is given that there exist real numbers $A, B$, and $C$ such that

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

for all $s \notin{p, q, r}$. What is $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$ ?
(A) 243
(B) 244
(C) 245
(D) 246
(E) 247

Answer:

(B) 244

Problem 25
For how many integers $n$ between 1 and 50 , inclusive, is

$$
\frac{\left(n^{2}-1\right)!}{(n!)^{n}}
$$

an integer? (Recall that $0!=1$.)
(A) 31
(B) 32
(C) 33
(D) 34
(E) 35

Answer:

(D) 34

American Mathematics Competition 10A - 2024

Problem 1
What is the value of $9901 \cdot 101-99 \cdot 10101$ ?
(A) 2
(B) 20
(C) 200
(D) 202
(E) 2020

Answer:

(A) 2

Problem 2
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=a L+b G$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take 69 minutes to hike to the top if a trail is 1.5 miles long and ascends 800 feet, as well as if a trail is 1.2 miles long and ascends 1100 feet. How many minutes does the model estimate it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet?
(A) 240
(B) 246
(C) 252
(D) 258
(E) 264

Answer:

(B) 246

Problem 3
What is the sum of the digits of the smallest prime that can be written as a sum of 5 distinct primes?
(A) 5
(B) 7
(C) 9
(D) 10
(E) 13

Answer:

(B) 7

Problem 4
The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
(A) 20
(B) 21
(C) 22
(D) 23
(E) 24

Answer:

(B) 21

Problem 5

What is the least value of $n$ such that $n!$ is a multiple of 2024 ?
(A) 11
(B) 21
(C) 22
(D) 23
(E) 253

Answer:

(D) 23

Problem 6
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA ?
(For example, 3 swaps are required to change ABC to CBA ; one such sequence of swaps is $\mathrm{ABC} \rightarrow \mathrm{BAC} \rightarrow \mathrm{BCA} \rightarrow \mathrm{CBA}$.)
(A) 6
(B) 10
(C) 12
(D) 15
(E) 24

Answer:

(D) 15

Problem 7
The product of three integers is 60 . What is the least possible positive sum of the three integers?
(A) 2
(B) 3
(C) 5
(D) 6
(E) 13

Answer:

(B) 3

Problem 8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00 PM and were able to pack 4, 3 , and 3 packages, respectively, every 3 minutes. At some later time, Daria joined the group, and Daria was able to pack 5 packages every 4 minutes. Together, they finished packing 450 packages at exactly 2:45 PM. At what time did Daria join the group?
(A) 1:25 PM
(B) 1:35 PM
(C) 1:45 PM
(D) 1:55 PM
(E) 2:05 PM

Answer:

(A) 1:25 PM

Problem 9
In how many ways can 6 juniors and 6 seniors form 3 disjoint teams of 4 people so that each team has 2 juniors and 2 seniors?
(A) 720
(B) 1350
(C) 2700
(D) 3280
(E) 8100

Answer:

(B) 1350

Problem 10
Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of 3 , then you replace $n$ by $\frac{n}{3}$. If $n$ is not a multiple of 3 , then you replace $n$ by $n+10$ . Then continue this process. For example, beginning with $n=4$, this procedure gives $\quad 4 \rightarrow 14 \rightarrow 24 \rightarrow 8 \rightarrow 18 \rightarrow 6 \rightarrow 2 \rightarrow 12 \rightarrow \cdots$. Suppose you start with $n=100$. What value results if you perform this operation exactly 100 times?
(A) 10
(B) 20
(C) 30
(D) 40
(E) 50

Answer:

Problem 11
How many ordered pairs of integers $(m, n)$ satisfy

$$
\sqrt{n^{2}-49}=m \text { ? }
$$

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

Answer:

(D) 4

Problem 12

Zelda played the Adventures of Math game on August 1 and scored 1700 points. She continued to play daily over the next 5 days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1700+80=1780$ points.) What was Zelda's average score in points over the 6 days?

(A) 1700
(B) 1702
(C) 1703
(D) 1713
(E) 1715

Answer:

(E) 1715

Problem 13

Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:

Of the 6 pairs of distinct transformations from this list, how many commute?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

Problem 14
One side of an equilateral triangle of height 24 lies on line $\ell$. A circle of radius 12 is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a \sqrt{b}-c \pi$, where $a, b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c$ ?
(A) 72
(B) 73
(C) 74
(D) 75
(E) 76

Answer:

(D) 75

Problem 15
Let $M$ be the greatest integer such that both $M+1213$ and $M+3773$ are perfect squares. What is the units digit of $M$ ?
(A) 1
(B) 2
(C) 3
(D) 6
(E) 8

Answer:

(E) 8

Problem 16

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $A B$ ?

(A) $4+4 \sqrt{5}$
(B) $10 \sqrt{2}$
(C) $5+5 \sqrt{5}$
(D) $10 \sqrt[4]{8}$
(E) 20

Answer:

(D) $10 \sqrt[4]{8}$

Problem 17
Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its\
probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m-\sqrt{n})$, where $m$ and $n$ are positive integers. What is $m+n$ ?
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

Answer:

(E) 14

Problem 18
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base$b$ integer $2024_{b}$ is divisible by 16 (where 16 is in base ten). What is the sum of the digits of $K$ ?
(A) 16
(B) 17
(C) 18
(D) 20
(E) 21

Answer:

(D) 20

Problem 19
The first three terms of a geometric sequence are the integers $a, 720$, and $b$, where $a<720<b$. What is the sum of the digits of the least possible value of $b$ ?
(A) 9
(B) 12
(C) 16
(D) 18
(E) 21

Answer:

(E) 21

Problem 20
Let $S$ be a subset of ${1,2,3, \ldots, 2024}$ such that the following two conditions hold:

What is the maximum possible number of elements in $S$ ?
(A) 436
(B) 506
(C) 608
(D) 654
(E) 675

Answer:

Problem 21
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length 5 . The numbers in positions $(5,5),(2,4),(4,3)$, and $(3,1)$ are $0,48,16$, and 12 , respectively. What number is in position $(1,2)$ ?

(A) 19
(B) 24
(C) 29
(D) 34
(E) 39

Answer:

Problem 22
Let $\mathcal{K}$ be the kite formed by joining two right triangles with legs 1 and $\sqrt{3}$ along a common hypotenuse. Eight copies of $\mathcal{K}$ are used to form the polygon shown below. What is the area of triangle $\triangle A B C$ ?

(A) $2+3 \sqrt{3}$
(B) $\frac{9}{2} \sqrt{3}$
(C) $\frac{10+8 \sqrt{3}}{3}$
(D) 8
(E) $5 \sqrt{3}$

Answer:

(B) $\frac{9}{2} \sqrt{3}$

Problem 23
The first three terms of a geometric sequence are the integers $a, 720$, and $b$, where $a<720<b$. What is the sum of the digits of the least possible value of $b$ ?
(A) 9
(B) 12
(C) 16
(D) 18
(E) 21

Answer:

(D) 18

Problem 24
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^{+}, A^{-}, B^{+}, B^{-}, C^{+}$, and $C^{-}$is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^{+}$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^{-}$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the
point $(0,0,0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
(A) $\frac{1}{54}$
(B) $\frac{7}{54}$
(C) $\frac{1}{6}$
(D) $\frac{5}{18}$
(E) $\frac{2}{5}$

Answer:

(B) $\frac{7}{54}$

Problem 25
The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of $1^{\prime \prime} \times 1^{\prime \prime}$ squares. Carl places 1 -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?

(A) 130
(B) 144
(C) 146
(D) 162
(E) 196

Answer:

American Mathematics Competition 12A - 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

Solution :

Let $h$ be the number of hours after Andy starts. Andy travels $8 h$ miles, and Betsy has traveled $12(h-1)$ miles since she started one hour later. Setting them equal:

$$
8 h=12(h-1) \Rightarrow 8 h=12 h-12 \Rightarrow 4 h=12 \Rightarrow h=3
$$

Since Andy started at 1:30, the catch-up time is 4:30. Answer: (E).
Alternatively, from Betsy's perspective: $8(h+1)=12 h \Rightarrow 8 h+8=12 h \Rightarrow h=2$ Same result: (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

Solution :

We are given $0.2(10)=2$ pounds of cashews in the first box.
Denote the pounds of nuts in the second nut mix as $x$.

$$
\begin{gathered}
5+0.2 x=0.4(10+x) \
0.2 x=1 \
x=5
\end{gathered}
$$

Thus, we have 5 pounds of the second mix.

$$
0.4(5)+2=2+2=\text { (B) } 4
$$

Problem 3

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

Solution :

When Ash joins a team, the team's average is pulled towards his age. Let $A$ be Ash's age and $N$ be the number of people on the student team. This means that there are $15-N$ people in the teacher team. Let us write an expression for the change in the average for each team.

The students originally had an average of 12 , which became 14 when Ash joined, so there was an increase of 2 . The term $A-12$ represents how much older Ash is compared to the average of the students'. If we divide this by $N+1$, which is the number of people on the student team when Ash joins, we get the average change per team member once Ash is added. Therefore,

$$
\frac{A-12}{N+1}=2 .
$$

Similarly, for teachers, the average was originally 55 , which decreased by 3 to become 52 when Ash joined. Intuitively, $55-A$ represents how much younger Ash is than the average age of the teachers. Dividing this by the expression $(15-N)+1$, which is the new total number of people on the teacher team, represents the average change per team member once Ash joins. We can write the equation

$$
\frac{55-A}{16-N}=3 .
$$

To solve the system, multiply equation (1) by $N+1$, and similarly multiply equation (2) by $16-N$. Then add the equations together, canceling $A$, leaving equation $43=50-N$. From this we get $N=7$ and $A=28$.

Problem 4:

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

At least one of these statements is true.
At least two of these statements are true.
At least two of these statements are false.
At least one of these statements is false.

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

Solution

We first number all the statements:
1) At least one of these statements is true. 2) At least two of these statements are true. 3) At least two of these statements are false. 4) At least one of these statements is false.

We can immediately see that statement 4 must be true, as it would contradict itself if it were false. Similarly, statement 1 must be true, as all the other statements must be false if it were false, which is contradictory because statement 4 is true. Since both 1 and 4 are true, statement 2 has to be true. Therefore, statement 3 is the only false statement, making the answer (B) 1.

Problem 5

Solution:

Problem 6:

Solution:

AMERICAN MATHEMATICS COMPETITION 8 - 2020

AMERICAN MATHEMATICS COMPETITION 10 A - 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

Answer:

Problem 16

The graph of $f(x)=|\lfloor x\rfloor|-|\lfloor 1-x\rfloor|$ is symmetric about which of the following? (Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
(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

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

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

Answer:

(D) 10

Problem 24

Answer:

(E) 20

Problem 25

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

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

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 - 2005

QUESTION 1 :

Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer?
(A) 7.5
(B) 15
(C) 30
(D) 120
(E) 240

ANSWER 1 :

(B) 15

QUESTION 2 :

Karl bought five folders from Pay-A-Lot at a cost of $\$ 2.50$ each. Pay-A-Lot had a $20 \%$-off sale the following day. How much could Karl have saved on the purchase by waiting a day?
(A) $\$ 1.00$
(B) $\$ 2.00$
(C) $\$ 2.50$
(D) $\$ 2.75$
(E) $\$ 5.00$

ANSWER 2 :

QUESTION 3 :

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

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

ANSWER 3 :

(D) 4

QUESTION 4 :

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1 \mathrm{~cm}, 8.2 \mathrm{~cm}$ and 9.7 cm . What is the area of the square in square centimeters?
(A) 24
(B) 25
(C) 36
(D) 48
(E) 64

ANSWER 4 :

QUESTION 5 :

Soda is sold in packs of 6,12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?
(A) 4
(B) 5
(C) 6
(D) 8
(E) 15

ANSWER 5 :

(B) 5

QUESTION 6:

Suppose $d$ is a digit. For how many values of $d$ is $2.00 d 5>2.005 ?$
(A) 0
(B) 4
(C) 5
(D) 6
(E) 10

ANSWER 6 :

QUESTION 7 :

Bill walks $\frac{1}{2}$ mile south, then $\frac{3}{4}$ mile east, and finally $\frac{1}{2}$ mile south. How many miles is he, in a direct line, from his starting point?
(A) 1
(B) $1 \frac{1}{4}$
(C) $1 \frac{1}{2}$
(D) $1 \frac{3}{4}$
(E) 2

ANSWER 7 :

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

QUESTION 8 :

Suppose $m$ and $n$ are positive odd integers. Which of the following must also be an odd integer?
(A) $m+3 n$
(B) $3 m-n$
(C) $3 m^2+3 n^2$
(D) $(n m+3)^2$
(E) $3 m n$

ANSWER 8 :

(E) $3 m n$

QUESTION 9 :

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

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

ANSWER 9 :

(D) 17

QUESTION 10 :

Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?
(A) 7
(B) 7.3
(C) 7.7
(D) 8
(E) 8.3

ANSWER 10 :

(D) 8

QUESTION 11 :

The sales tax rate in Rubenenkoville is $6 \%$. During a sale at the Bergville Coat Closet, the price of a coat is discounted $20 \%$ from its $\$ 90.00$ price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $\$ 90.00$ and adds $6 \%$ sales tax, then subtracts $20 \%$ from this total. Jill rings up $\$ 90.00$, subtracts $20 \%$ of the price, then adds $6 \%$ of the discounted price for sales tax. What is Jack's total minus Jill's total?
(A) $-\$ 1.06$
(B) $-\$ 0.53$
(C) $\$ 0$
(D) $\$ 0.53$
(E) $\$ 1.06$

ANSWER 11 :

QUESTION 12 :

Big AI, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5?
(A) 20
(B) 22
(C) 30
(D) 32
(E) 34

ANSWER 12 :

(D) 32

QUESTION 13 :

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

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

ANSWER 13 :

QUESTION 14 :

The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
(A) 80
(B) 96
(C) 100
(D) 108
(E) 192

ANSWER 14 :

(B) 96

QUESTION 15 :

How many different isosceles triangles have integer side lengths and perimeter $23 ?$
(A) 2
(B) 4
(C) 6
(D) 9
(E) 11

ANSWER 15 :

QUESTION 16 :

A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
(A) 6
(B) 9
(C) 12
(D) 13
(E) 15

ANSWER 16 :

(D) 13

QUESTION 17 :

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?

(A) Angela
(B) Briana
(C) Carla
(D) Debra
(E) Evelyn

ANSWER 17 :

(E) Evelyn

QUESTION 18 :

How many three-digit numbers are divisible by 13 ?
(A) 7
(B) 67
(C) 69
(D) 76
(E) 77

ANSWER 18 :

QUESTION 19 :

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

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

ANSWER 19 :

(A) 180

QUESTION 20 :

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12 . Both start on point 12 . Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
(A) 6
(B) 8
(C) 12
(D) 14
(E) 24

ANSWER 20 :

(A) 6

QUESTION 21 :

How many distinct triangles can be drawn using three of the dots below as vertices?

(A) 9
(B) 12
(C) 18
(D) 20
(E) 24

ANSWER 21 :

QUESTION 22 :

A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50\% more than the small size and contains $20 \%$ less detergent than the large size. The large size contains twice as much detergent as the small size and costs $30 \%$ more than the medium size. Rank the three sizes from best to worst buy.
(A) $S M L$
(B) $L M S$
(C) $M S L$
(D) $L S M$
(E) $M L S$

ANSWER 22 :

(E) $M L S$

QUESTION 23 :

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

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

ANSWER 23 :

(B) 8

QUESTION 24 :

A certain calculator has only two keys $[+1]$ and $[\mathrm{x} 2]$. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed " 9 " and you pressed [ +1 ], it would display " 10 ." If you then pressed [ $x 2$ ], it would display " 20 ." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

ANSWER 24 :

(B) 9

QUESTION 25 :

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

ANSWER 25 :

American Mathematics Competition 8 - 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

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

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

Answer:

Problem 7

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

Answer:

(E) 149

Problem 12

Answer:

(D) 464

Problem 13

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

Answer:

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

Problem 15

(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

Amswer:

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

Problem 17

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

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

Answer:

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

Problem 19

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

Answer:

(A) 119

Problem 21

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

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

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

Answer:

(A) 25