American Mathematics Competition 10A - 2021

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

Answer:

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

Answer:

(E) 20

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

Answer:

(D) 6

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

Answer:

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

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

Answer:

(E) 9

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

Answer:

(B) 8

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

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

Answer:

(D) 170

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

Answer:

(B) 1

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

Answer:

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

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

Answer:

(B) -13.5

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

Answer:

(A) 70

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

Answer:

(D) 3

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

Answer:

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

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

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

Answer:

(D) 5

Problem 15

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

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

Answer:

Problem 16

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

Answer:

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

Problem 17

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

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

Answer:

(D) 17

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

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

Answer:

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

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

Answer:

(A) 10

Problem 20

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

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

Answer:

(B) 6

Problem 21

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

Answer:

(E) 16

Problem 22

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

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

Answer:

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

Problem 23

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

Answer:

(D) 10

Problem 24

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

Answer:

(E) 20

Problem 25

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

Answer:

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

AMERICAN MATHEMATICS COMPETITION 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

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

Amswer:

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

(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 - 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 - 2004

QUESTION 1 :

On a map, a 12 -centimeter length represents 72 kilometers. How many kilometers does a 17 -centimeter length represent?
(A) 6
(B) 102
(C) 204
(D) 864
(E) 1224

ANSWER 1 :

(B) 102

QUESTION 2 :

How many different four-digit numbers can be formed by rearranging the four digits in 2004 ?
(A) 4
(B) 6
(C) 16
(D) 24
(E) 81

ANSWER 2 :

(B) 6

QUESTION 3 :

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they share, how many meals should they have ordered to have just enough food for the 12 of them?
(A) 8
(B) 9
(C) 10
(D) 15
(E) 18

ANSWER 3 :

(A) 8

QUESTION 4 :

Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
(A) 2
(B) 4
(C) 6
(D) 8
(E) 10

ANSWER 4 :

(B) 4

QUESTION 5 :

Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
(A) 4
(B) 7
(C) 8
(D) 15
(E) 16

ANSWER 5 :

(D) 15

QUESTION 6 :

After Sally takes 20 shots, she has made $55 \%$ of her shots. After she takes 5 more shots, she raises her percentage to $56 \%$. How many of the last 5 shots did she make?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER 6:

QUESTION 7 :

An athlete's target heart rate, in beats per minute, is $80 \%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220 . To the nearest whole number, what is the target heart rate of an athlete
who is 26 years old?
(A) 134
(B) 155
(C) 176
(D) 194
(E) 243

ANSWER 7:

(B) 155

QUESTION 8 :

Find the number of two-digit positive integers whose digits total 7.
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

ANSWER 8 :

(B) 7

QUESTION 9 :

The average of the five numbers in a list is 54 . The average of the first two numbers is 48 . What is the average of the last three numbers?
(A) 55
(B) 56
(C) 57
(D) 58
(E) 59

ANSWER 9 :

(D) 58

QUESTION 10 :

Handy Aaron helped a neighbor $1 \frac{1}{4}$ hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\$ 3$ per hour. How much did he earn for the week?
(A) $\$ 8$
(B) $\$ 9$
(C) $\$ 10$
(D) $\$ 12$
(E) $\$ 15$

ANSWER 10:

(E) $\$ 15$

QUESTION 11:

The numbers $-2,4,6,9$ and 12 are rearranged according to these rules:

The largest isn't first, but it is in one of the first three places.
The smallest isn't last, but it is in one of the last three places.
The median isn't first or last.

What is the average of the first and last numbers?
(A) 3.5
(B) 5
(C) 6.5
(D) 7.5
(E) 8

ANSWER 11 :

QUESTION 12 :

Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during
that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
(A) 7
(B) 8
(C) 11
(D) 14
(E) 15

ANSWER 12 :

(B) 8

QUESTION 13 :

Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true.
I. Bill is the oldest.
II. Amy is not the oldest.
III. Celine is not the youngest.

Rank the friends from the oldest to the youngest.
(A) Bill, Amy, Celine
(B) Amy, Bill, Celine
(C) Celine, Amy, Bill
(D) Celine, Bill, Amy
(E) Amy, Celine, Bill

ANSWER 13 :

(E) Amy, Celine, Bill

QUESTION 14 :

What is the area enclosed by the geoboard quadrilateral below?

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

ANSWER 14:

QUESTION 15 :

Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?

(A) 5
(B) 7
(C) 11
(D) 12
(E) 18

ANSWER 15 :

QUESTION 16 :

Two 600 mL pitchers contain orange juice. One pitcher is $1 / 3$ full and the other pitcher is $2 / 5$ full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
(A) $\frac{1}{8}$
(B) $\frac{3}{16}$
(C) $\frac{11}{30}$
(D) $\frac{11}{19}$
(E) $\frac{11}{15}$

ANSWER 16 :

QUESTION 17 :

Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?
(A) 1
(B) 3
(C) 6
(D) 10
(E) 12

ANSWER 17 :

(D) 10

QUESTION 18 :

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10 . Each throw hits the target in a region with a different value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points?
(A) Alice
(B) Ben
(C) Cindy
(D) Dave
(E) Ellen

ANSWER 18 :

(A) Alice

QUESTION 19 :

A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers $3,4,5$, and 6 . The smallest such number lies between which two numbers?
(A) 40 and 49
(B) 60 and 79
(C) 100 and 129
(D) 210 and 249
(E) 320 and 369

ANSWER 19 :

(B) 60 and 79

QUESTION 20 :

Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
(A) 12
(B) 18
(C) 24
(D) 27
(E) 36

ANSWER 20 :

(D) 27

QUESTION 21 :

Spinners $A$ and $B$ are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?

ANSWER 21 :

QUESTION 22 :

At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men?
(A) $\frac{1}{3}$
(B) $\frac{3}{8}$
(C) $\frac{2}{5}$
(D) $\frac{5}{12}$
(E) $\frac{3}{5}$

ANSWER 22 :

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

QUESTION 23 :

Tess runs counterclockwise around rectangular block $J K L M$. She lives at corner $J$. Which graph could represent her straight-line distance from home?

ANSWER 23 :

QUESTION 24 :

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

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

ANSWER 24 :

QUESTION 25 :

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

ANSWER 26 :

AMERICAN MATHEMATICS COMPETITION 8 - 2007

Problem 1
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks, she helps around the house for $8,11,7,12$ and 10 hours. How many hours must she work during the final week to earn the tickets?
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

Answer:

(D) 12

Problem 2
Six-hundred fifty students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

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

Answer:

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

Problem 3
What is the sum of the two smallest prime factors of 250 ?
(A) 2
(B) 5
(C) 7
(D) 10
(E) 12

Answer:

Problem 4
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?
(A) 12
(B) 15
(C) 18
(D) 30
(E) 36

Answer:

(D) 30

Problem 5
Chandler wants to buy a $\$ 500$ dollar mountain bike. For his birthday, his grandparents send him $\$ 50$, his aunt sends him $\$ 35$ and his cousin gives him $\$ 15$. He earns $\$ 16$ per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

Answer:

(B) 25

Problem 6

The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
(A) 7
(B) 17
(C) 34
(D) 41
(E) 80

Answer:

(E) 80

Problem 7

The average age of 5 people in a room is 30 years. An 18-year-old person leaves the room. What is the average age of the four remaining people?
(A) 25
(B) 26
(C) 29
(D) 33
(E) 36

Answer:

(D) 33

Problem 8
In trapezoid $A B C D, A D$ is perpendicular to $D C, A D=A B=3$, and $D C=6$. In addition, E is on $D C$, and $B E$ is parallel to $A D$. Find the area of $\triangle B E C$.

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

Answer:

(B) 4.5

Problem 9
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square?

(A) 1
(B) 2
(C) 3
(D) 4
(E) cannot be determined

Answer:

(B) 2

Problem 10

For any positive integer $n$, define $n$ to be the sum of the positive factors of $n$. For example, $6=1+2+3+6=12$.\
Find 11 .
(A) 13
(B) 20
(C) 24
(D) 28
(E) 30

Answer:

(D) 28

Problem 11
Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$ ?

(A) $I$
(B) $I I$
(C) III
(D) $I V$
(E) cannot be determined

Answer:

(D) $I V$

Problem 12

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

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

Answer:

(A) $1: 1$

Problem 13
Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

(A) 503
(B) 1006
(C) 1504
(D) 1507
(E) 1510

Answer:

Problem 14
The base of isosceles $\triangle A B C$ is 24 and its area is 60 . What is the length of one of the congruent sides?
(A) 5
(B) 8
(C) 13
(D) 14
(E) 18

Answer:

Problem 15
Let $a, b$ and $c$ be numbers with $0<a<b<c$. Which of the following is impossible?
(A) $a+c<b$
(B) $a \cdot b<c$
(C) $a+b<c$
(D) $a \cdot c<b$
(E) $\frac{b}{c}=a$

Answer:

(A) $a+c<b$

Problem 16
Amanda Reckonwith draws five circles with radii $1,2,3,4$ and 5 . Then for each circle she plots the point ( $C ; A$ ), where $C$ is its circumference and $A$ is its area. Which of the following could be her graph?

Answer:

Problem 17

A mixture of 30 liters of paint is $25 \%$ red tint, $30 \%$ yellow tint, and $45 \%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture?

Answer:

Problem 18
The product of the two 99 -digit numbers
$303,030,303, \ldots, 030,303$ and $505,050,505, \ldots, 050,505$ has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$ ?
(A) 3
(B) 5
(C) 6
(D) 8
(E) 10

Answer:

(D) 8

Problem 19
Pick two consecutive positive integers whose sum is less than 100 . Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
(A) 2
(B) 64
(C) 79
(D) 96
(E) 131

Answer:

Problem 20

Before district play, the Unicorns had won $45 \%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?

Answer:

(A) 48

Problem 21

Two cards are dealt from a deck of four red cards labeled $A, B, C, D$ and four green cards labeled $A, B, C, D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
(A) $\frac{2}{7}$
(B) $\frac{3}{8}$
(C) $\frac{1}{2}$
(D) $\frac{4}{7}$
(E) $\frac{5}{8}$

Answer:

(D) $\frac{4}{7}$

Problem 22
A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
(A) 2
(B) 4.5
(C) 5
(D) 6.2
(E) 7

Answer:

Problem 23
What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?

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

Answer:

(B) 6

Problem 24

A bag contains four pieces of paper, each labeled with one of the digits $1,2,3$ or 4 , with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3 ?
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$

Answer:

Porblem 25

On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given
region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

(A) $\frac{17}{36}$
(B) $\frac{35}{72}$
(C) $\frac{1}{2}$
(D) $\frac{37}{72}$
(E) $\frac{19}{36}$

Answer:

(B) $\frac{35}{72}$

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 8 - 2015

Question 1 :

Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is 12 feet long and 9 feet wide? (There are 3 feet in a yard.)
(A) 12
(B) 36
(C) 108
(D) 324
(E) 972

Answer 1 :

(A) 12

Question 2 :

Point $O$ is the center of the regular octagon $A B C D E F G H$, and $X$ is the midpoint of the side $\overline{A B}$. What fraction of the area of the octagon is shaded?
(A) $\frac{11}{32}$
(B) $\frac{3}{8}$
(C) $\frac{13}{32}$
(D) $\frac{7}{16}$
(E) $\frac{15}{32}$

Answer 2 :

Question 3 :

Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of 10 miles per hour. Jack walks to the pool at a constant speed of 4 miles per hour. How many minutes before Jack does Jill arrive?
(A) 5
(B) 6
(C) 8
(D) 9
(E) 10

Answer 3 :

(D) 9

Question 4 :

The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
(A) 2
(B) 4
(C) 5
(D) 6
(E) 12

Answer 4 :

(E) 12

Question 5 :

Billy's basketball team scored the following points over the course of the first 11 games of the season. If his team scores 40 in the $12^{\text {th }}$ game, which of the following statistics will show an increase?

$$
42,47,53,53,58,58,58,61,64,65,73
$$

(A) range
(B) median
(C) mean
(D) mode
(E) mid-range

Answer 5 :

(A) range

Question 6 :

In $\triangle A B C, A B=B C=29$, and $A C=42$. What is the area of $\triangle A B C$ ?
(A) 100
(B) 420
(C) 500
(D) 609
(E) 701

Answer 6 :

(B) 420

Question 7 :

Each of two boxes contains three chips numbered $1,2,3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
(A) $\frac{1}{9}$
(B) $\frac{2}{9}$
(C) $\frac{4}{9}$
(D) $\frac{1}{2}$
(E) $\frac{5}{9}$

Answer 7 :

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

Question 8 :

What is the smallest whole number larger than the perimeter of any triangle with a side of length 5 and a side of length $19 ?$
(A) 24
(B) 29
(C) 43
(D) 48
(E) 57

Answer 8 :

(D) 48

Question 9 :

On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?
(A) 39
(B) 40
(C) 210
(D) 400
(E) 401

Answer 9 :

(D) 400

Question 10 :

How many integers between 1000 and 9999 have four distinct digits?
(A) 3024
(B) 4536
(C) 5040
(D) 6480
(E) 6561

Answer 10 :

(B) 4536

Question 11 :

In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit ( 0 through 9 ). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
(A) $\frac{1}{22,050}$
(B) $\frac{1}{21,000}$
(C) $\frac{1}{10,500}$
(D) $\frac{1}{2,100}$
(E) $\frac{1}{1,050}$

Answer 11 :

(B) $\frac{1}{21,000}$

Question 12 :

How many pairs of parallel edges, such as $\overline{A B}$ and $\overline{G H}$ or $\overline{E H}$ and $\overline{F G}$, does a cube have?

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

Answer 12 :

Question 13 :

How many subsets of two elements can be removed from the set ${1,2,3,4,5,6,7,8,9,10,11}$ so that the mean (average) of the remaining numbers is 6 ?
(A) 1
(B) 2
(C) 3
(D) 5
(E) 6

Answer 13 :

(D) 5

Question 14 :

Which of the following integers cannot be written as the sum of four consecutive odd integers?
(A) 16
(B) 40
(C) 72
(D) 100
(E) 200

Answer 14 :

(D) 100

Question 15 :

At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?
(A) 49
(B) 70
(C) 79
(D) 99
(E) 149

Answer 15 :

(D) 99

Question 16 :

In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\frac{1}{3}$ of all the ninth graders are paired with $\frac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
(A) $\frac{2}{15}$
(B) $\frac{4}{11}$
(C) $\frac{11}{30}$
(D) $\frac{3}{8}$
(E) $\frac{11}{15}$

Answer 16 :

(B) $\frac{4}{11}$

Question 17 :

Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?
(A) 4
(B) 6
(C) 8
(D) 9
(E) 12

Answer 17 :

(D) 9

Question 18 :

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is 2 and the constant added is 3 . Each row and each column in this $5 \times 5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$ ?
(A) 21
(B) 31
(C) 36
(D) 40
(E) 42

Answer 18 :

(B) 31

Question 19 :

A triangle with vertices as $A=(1,3), B=(5,1)$, and $C=(4,4)$ is plotted on a $6 \times 5$ grid. What fraction of the grid is covered by the triangle?
(A) $\frac{1}{6}$
(B) $\frac{1}{5}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$

Answer 19 :

(A) $\frac{1}{6}$

Question 20 :

Ralph went to the store and bought 12 pairs of socks for a total of $\$ 24$. Some of the socks he bought cost $\$ 1$ a pair, some of the socks he bought cost $\$ 3$ a pair, and some of the socks he bought cost $\$ 4$ a pair. If he bought at least one pair of each type, how many pairs of $\$ 1$ socks did Ralph buy?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Answer 20 :

(D) 7

Question 21 :

In the given figure, hexagon $A B C D E F$ is equiangular, $A B J I$ and $F E H G$ are squares with areas 18 and 32 respectively, $\triangle J B K$ is equilateral and $F E=B C$. What is the area of $\triangle K B C$ ?

Answer 21 :

Question 22 :

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
(A) 21
(B) 30
(C) 60
(D) 90
(E) 1080

Answer 22 :

Question 23 :

Tom has twelve slips of paper which he wants to put into five cups labeled $A, B, C, D, E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2,2,2,2.5,2.5,3,3,3,3,3.5,4$, and 4.5 . If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup?
(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) $E$

Answer 23 :

(D) $D$

Question 24 :

A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2 M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division?
(A) 36
(B) 48
(C) 54
(D) 60
(E) 72

Answer 24 :

(B) 48

Question 25 :

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

Answer 25 :

American Mathematics Competition - 2016

Question 1 :

The longest professional tennis match ever played lasted a total of 11 hours and 5 minutes. How many minutes was this?
(A) 605
(B) 655
(C) 665
(D) 1005
(E) 1105

Answer 1 :

Question 2 :

In rectangle $A B C D, A B=6$ and $A D=8$. Point $M$ is the midpoint of $\overline{A D}$. What is the area of $\triangle A M C$ ?

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

Answer 2 :

(A) 12

Question 3 :

Four students take an exam. Three of their scores are 70,80 , and 90 . If the average of their four scores is 70 , then what is the remaining score?
(A) 40
(B) 50
(C) 55
(D) 60
(E) 70

Answer 3 :

(A) 40

Question 4 :

When Cheenu was a boy, he could run 15 miles in 3 hours and 30 minutes. As an old man, he can now walk 10 miles in 4 hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?
(A) 6
(B) 10
(C) 15
(D) 18
(E) 30

Answer 4 :

(B) 10

Question 5 :

The number $N$ is a two-digit number.

When $N$ is divided by 9 , the remainder is 1 .
When $N$ is divided by 10 , the remainder is 3 .

What is the remainder when $N$ is divided by 11 ?
(A) 0
(B) 2
(C) 4
(D) 5
(E) 7

Answer 5 :

(E) 7

Question 6 :

The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?

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

Answer 6 :

(B) 4

Question 7 :

Which of the following numbers is not a perfect square?
(A) $1^{2016}$
(B) $2^{2017}$
(C) $3^{2018}$
(D) $4^{2019}$
(E) $5^{2020}$

Answer 7 :

(B) $2^{2017}$

Question 8 :

Find the value of the expression

$$
100-98+96-94+92-90+\cdots+8-6+4-2 .
$$

(A) 20
(B) 40
(C) 50
(D) 80
(E) 100

Answer 8 :

Question 9 :

What is the sum of the distinct prime integer divisors of $2016 ?$
(A) 9
(B) 12
(C) 16
(D) 49
(E) 63

Answer 9 :

(B) 12

Question 10 :

Suppose that $a * b$ means $3 a-b$. What is the value of $x$ if

$$
2 *(5 * x)=1
$$

(A) $\frac{1}{10}$
(B) 2
(C) $\frac{10}{3}$
(D) 10
(E) 14

Answer 10 :

(D) 10

Question 11 :

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is 132 .
(A) 5
(B) 7
(C) 9
(D) 11
(E) 12

Answer 11 :

(B) 7

Question 12 :

Jefferson Middle School has the same number of boys and girls. $\frac{3}{4}$ of the girls and $\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls?
(A) $\frac{1}{2}$
(B) $\frac{9}{17}$
(C) $\frac{7}{13}$
(D) $\frac{2}{3}$
(E) $\frac{14}{15}$

Answer 12 :

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

Question 13 :

Two different numbers are randomly selected from the set ${-2,-1,0,3,4,5}$ and multiplied together. What is the probability that the product is 0 ?
(A) $\frac{1}{6}$
(B) $\frac{1}{5}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{1}{2}$

Answer 13 :

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

Question 14 :

Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?
(A) 525
(B) 560
(C) 595
(D) 665
(E) 735

Answer 14 :

(A) 525

Question 15 :

What is the largest power of 2 that is a divisor of $13^4-11^4 ?$
(A) 8
(B) 16
(C) 32
(D) 64
(E) 128

Answer 15 :

Question 16 :

Annie and Bonnie are running laps around a 400 -meter oval track. They started together, but Annie has pulled ahead, because she runs $25 \%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
(A) $1 \frac{1}{4}$
(B) $3 \frac{1}{3}$
(C) 4
(D) 5
(E) 25

Answer 16 :

(D) 5

Question 17 :

An ATM password at Fred's Bank is composed of four digits from 0 to 9 , with repeated digits allowable. If no password may begin with the sequence $9,1,1$, then how many passwords are possible?
(A) 30
(B) 7290
(C) 9000
(D) 9990
(E) 9999

Answer 17 :

(D) 9990

Question 18 :

In an All-Area track meet, 216 sprinters enter a 100 - meter dash competition. The track has 6 lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
(A) 36
(B) 42
(C) 43
(D) 60
(E) 72

Answer 18 :

Question 19 :

The sum of 25 consecutive even integers is 10,000 . What is the largest of these 25 consecutive integers?
(A) 360
(B) 388
(C) 412
(D) 416
(E) 424

Answer 19 :

(E) 424

Question 20 :

The least common multiple of $a$ and $b$ is 12 , and the least common multiple of $b$ and $c$ is 15 . What is the least possible value of the least common multiple of $a$ and $c$ ?
(A) 20
(B) 30
(C) 60
(D) 120
(E) 180

Answer 20 :

(A) 20

Question 21 :

A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
(A) $\frac{3}{10}$
(B) $\frac{2}{5}$
(C) $\frac{1}{2}$
(D) $\frac{3}{5}$
(E) $\frac{7}{10}$

Answer 21 :

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

Question 22 :

Rectangle $D E F A$ below is a $3 \times 4$ rectangle with $D C=C B=B A=1$. The area of the "bat wings" (shaded area) is

Answer 22 :

Question 23 :

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle C E D$ ?
(A) 90
(B) 105
(C) 120
(D) 135
(E) 150

Answer 23 :

Question 24 :

The digits $1,2,3,4$, and 5 are each used once to write a five-digit number $P Q R S T$. The three-digit number $P Q R$ is divisible by 4 , the threedigit number $Q R S$ is divisible by 5 , and the three-digit number $R S T$ is divisible by 3 . What is $P$ ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer 24 :

(A) 1

Question 25 :

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

Answer 25 :