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 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 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

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? (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

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 - 2025

Problem 1

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

American Mathematics Competition 8 - 2024

Problem 1

What is the ones digit of

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

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

Answer:

(B) 2

Problem 2

What is the value of this expression in decimal form?

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

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

Answer:

Problem 3

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

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

Answer:

(E) 52

Problem 4

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

Answer:

(E) 9

Problem 5

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

Answer:

(B) 6

Problem 6

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

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

Answer:

(D) R, P, S, Q

Problem 7

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

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

Answer:

(E) 5

Problem 8

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

Answer:

(D) 6

Problem 9

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

Answer:

(E) 28

Problem 10

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

Answer:

(B) 414

Problem 11

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

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

Answer:

(D) 11

Problem 12

Rohan keeps a total of 90 guppies in 4 fish tanks.

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

Answer:

(E) 26

Problem 13

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

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

Answer:

(B) 5

Problem 14

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

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

Answer:

(A) 28

Problem 15

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

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

Answer:

Problem 16

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

Answer:

(D) 11

Problem 17

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

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

Answer:

(E) 32

Problem 18

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

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

Answer:

(A) 108

Problem 19

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

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

Answer:

Problem 20

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

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

Answer:

(D) 3

Problem 21

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

Answer:

(E) 24

Problem 22

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

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

Answer:

(B) 600

Problem 23

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

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

Answer:

Problem 24

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

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

Answer:

(B) 5

Problem 25

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

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

Answer:

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:

American Mathematics Competition - 2011

Problem 1

Margie bought 3 apples at a cost of 50 cents each. She paid with a 5 -dollar bill. How much change did Margie receive?

Answer:

(E) Is the correct answer.

Problem 2

Karl's rectangular vegetable garden is 20 by 45 feet, and Makenna's is 25 by 40 feet. Which garden is larger in area?

(A) Karl's garden is larger by 100 square feet.

(B) Karl's garden is larger by 25 square feet.

(D) Makenna's garden is larger by 25 square feet.

(E) Makenna's garden is larger by 100 square feet.

Answer:

(E) Makenna's garden is larger by 100 square feet.

Problem 3

Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?

Answer:

(D) Is the correct answer.

Problem 4

Here is a list of the numbers of fish that Tyler caught in nine outings last summer:

Which statement about the mean, median, and mode is true?

Answer:

Problem 5

What time was it 2011 minutes after midnight on January 1, 2011?

(A)January 1 at 9:31PM

(B)January 1 at 11:51PM

(C)January 2 at 3:11AM

(D)January 2 at 9:31AM

(E)January 2 at 6:01PM

Answer:

(D)January 2 at 9:31AM

Problem 6

In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?
(A) 20
(B) 25
(C) 45
(D)306
(E)351

Answer:

(D)306

Problem 7

Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?

Answer:

Problem 8

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4 , and 6 . If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 9

Answer:

(B) 5

Problem 9

Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?

(A) 2
(B) 2.5
(C) 4
(D) 4.5
(E) 5

Answer:

(E) 5

Problem 10

The taxi fare in Gotham City is $\$ 2.40$ for the first $\frac{1}{2}$ mile and additional mileage charged at the rate $\$ 0.20$ for each additional 0.1 mile. You plan to give the driver a $\$ 2$ tip. How many miles can you ride for $\$ 10$ ?
(A) 3.0
(B) 3.25
(C) 3.3
(D) 3.5
(E) 3.75

Answer:

Problem 11

The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?

(A) 6
(B) 8
(C) 9
(D) 10
(E) 12

Answer:

(A) 6

Problem 12

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

Answer:

(B) Is the correct answer.

Problem 13

Two congruent squares, $A B C D$ and $P Q R S$, have side length 15. They overlap to form the 15 by 25 rectangle $A Q R D$ shown. What percent of the area of rectangle $A Q R D$ is shaded?

(A) 15
(B) 18
(C) 20
(D) 24
(E) 25

Answer:

Problem 14

There are 270 students at Colfax Middle School, where the ratio of boys to girls is $5: 4$. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is $4: 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

Answer:

Problem 15

How many digits are in the product $4^{5} \cdot 5^{10}$ ?

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

Answer:

(D) 11

Problem 16

Let $A$ be the area of the triangle with sides of length 25,25 , and 30 . Let $B$ be the area of the triangle with sides of length 25,25 , and 40 . What is the relationship between $A$ and $B$ ?

Answer:

Problem 17

Let $w, x, y$, and $z$ be whole numbers. If $2^{w} \cdot 3^{x} \cdot 5^{y} \cdot 7^{z}=588$, then what does $2 w+3 x+5 y+7 z$ equal?

(A) 21
(B) 25
(C) 27
(D) 35
(E) 56

Answer:

(A) 21

Problem 18

A fair 6 -sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?

Answer:

(D) Is the correct answer.

Problem 19

How many rectangles are in this figure?

(A) 8

(B) 9

(D) 11

(E) 12

Answer:

(D) 11

Problem 20

Quadrilateral $A B C D$ is a trapezoid, $A D=15, A B=50, B C=20$, and the altitude is 12 . What is the area of the trapeziod?

Answer:

(D) Is the correct answer.

Problem 21

Students guess that Norb's age is $24,28,30,32,36,38,41,44,47$, and 49 . Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?

(A) 29
(B)31
(C) 37
(D)43
(E) 48

Answer:

Problem 22

22 What is the tens digit of $7^{2011}$ ?

(A) 0
(B) 1
(C) 3
(D) 4
(E) 7

Answer:

(D) 4

Problem 23

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5 , and 5 is the largest digit?

(A) 24
(B) 48
(C) 60
(D) 84
(E) 108

Answer:

(D) 84

Problem 24

In how many ways can 10001 be written as the sum of two primes?

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

Answer:

(A) 0

Problem 25

A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?

Answer:

(A) Is the correct answer.