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 - 2000

PROBLEM 1 :

Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
(A) 15
(B) 16
(C) 17
(D) 21
(E) 37

ANSWER : (B) 16

PROBLEM 2 :

Which of these numbers is less than its reciprocal?
(A) -2
(B) -1
(C) 0
(D) 1
(E) 2

ANSWER : (A) -2

PROBLEM 3 :

How many whole numbers lie in the interval between $\frac{5}{3}$ and $2 \pi$ ?
(A) 2
(B) 3
(C) 4
(D) 5
(E) infinitely many

ANSWER : (D) 5

PROBLEM 4 :

In 1960 only $5 \%$ of the working adults in Carlin City worked at home. By 1970 the "at-home" work force had increased to $8 \%$. In 1980 there were approximately $15 \%$ working at home, and in 1990 there were $30 \%$. The graph that best illustrates this is:

ANSWER : (E)

PROBLEM 5 :

Each principal of Lincoln High School serves exactly one 3 -year term. What is the maximum number of principals this school could have during an 8 -year period?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 8

ANSWER : (C) 4

PROBLEM 6 :

Figure $A B C D$ is a square. Inside this square three smaller squares are drawn with side lengths as labeled. the area of the shaded L-shaped region is

(A) 7
(B) 10
(C) 12.5
(D) 14
(E) 15

ANSWER :(A) 7

PROBLEM 7 :

What is the minimum possible product of three different numbers of the set ${-8,-6,-4,0,3,5,7}$ ?
(A) -336
(B) -280
(C) -210
(D) -192
(E) 0

ANSWER : (B) -280

PROBLEM 8 :

Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). The total number of dots NOT visible in this view is

(A) 21
(B) 22
(C) 31
(D) 41
(E) 53

ANSWER : (D) 41

PROBLEM 9 :

Three-digit powers of 2 and 5 are used in this cross-number puzzle. What is the only possible digit for the outlined square?

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

ANSWER : (D)

PROBLEM 10 :

Ara and Shea were once the same height. Since then Shea has grown $20 \%$ while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?
(A) 48
(B) 51
(C) 52
(D) 54
(E) 55

ANSWER : (E) 55

PROBLEM 11 :

The number 64 has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property?
(A) 15
(B) 16
(C) 17
(D) 18
(E) 20

ANSWER : (C) 17

PROBLEM 12 :

A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?

(A) 344
(B) 347
(C) 350
(D) 353
(E) 356

ANSWER : (D) 353

PROBLEM 13 :

In triangle $C A T$, we have $\angle A C T=\angle A T C$ and $\angle C A T=36^{\circ}$. If $\overline{T R}$ bisects $\angle A T C$, then $\angle C R T=$

(A) $16^{\circ}$
(B) $51^{\circ}$
(C) $72^{\circ}$
(D) $90^{\circ}$
(E) $108^{\circ}$

ANSWER : (C) $72^{\circ}$

PROBLEM 14 :

What is the units digit of $19^{19}+99^{99}$ ?
(A) 0
(B) 1
(C) 2
(D) 8
(E) 9

ANSWER : (D) 8

PROBLEM 15 :

Triangle $A B C, A D E$, and $E F G$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{A C}$ and $\overline{A E}$, respectively. If $A B=4$, what is the perimeter of figure $A B C D E F G$ ?

(A) 12
(B) 13
(C) 15
(D) 18
(E) 21

ANSWER : (C) 15

PROBLEM 16 :

In order for Mateen to walk a kilometer $(1000 \mathrm{~m})$ in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
(A) 40
(B) 200
(C) 400
(D) 500
(E) 1000

ANSWER : (C) 400

PROBLEM 17 :

The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b=\frac{a^{2}}{b}$. Determine $[(1 \otimes 2) \otimes 3]-[1 \otimes(2 \otimes 3)]$.
(A) $-\frac{2}{3}$
(B) $-\frac{1}{4}$
(C) 0
(D) $\frac{1}{4}$
(E) $\frac{2}{3}$

ANSWER : (A) $-\frac{2}{3}$

PROBLEM 18 :

Consider these two geoboard quadrilaterals. Which of the following statements is true?

(A) The area of quadrilateral I is more than the area of quadri- • lateral II.
(B) The area of quadrilateral I is less than the area of quadrilateral II.
(C) The quadrilaterals have the same area and the same perimeter.
(D) The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.
(E) The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.

ANSWER : (E) The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.

PROBLEM 19 :

Three circular arcs of radius 5 units bound the region shown. $\operatorname{Arcs} A B$ and $A D$ are quarter-circles, and $\operatorname{arc} B C D$ is a semicircle. What is the area, in square units, of the region?

(A) 25
(B) $10+5 \pi$
(C) 50
(D) $50+5 \pi$
(E) $25 \pi$

ANSWER : (C) 50

PROBLEM 20 :

You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $\$ 1.02$, with at least one coin of each type. How many dimes must you have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER : (A) 1

PROBLEM 21 :

Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is
(A) $\frac{1}{4}$
(B) $\frac{3}{8}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{3}{4}$

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

PROBLEM 22 :

A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to:

(A) 10
(B) 15
(C) 17
(D) 21
(E) 25

ANSWER : (C) 17

PROBLEM 23 :

There is a list of seven numbers. The average of the first four numbers is 5 , and the average of the last four numbers is 8 . If the average of all seven numbers is $6 \frac{4}{7}$, then the number common to both sets of four numbers is
(A) $5 \frac{3}{7}$
(B) 6
(C) $6 \frac{4}{7}$
(D) 7
(E) $7 \frac{3}{7}$

ANSWER : (B) 6

PROBLEM 24 :

If $\angle A=20^{\circ}$ and $\angle A F G=\angle A G F$, Then $\angle B+\angle D=$

(A) $48^{\circ}$
(B) $60^{\circ}$
(C) $72^{\circ}$
(D) $80^{\circ}$
(E) $90^{\circ}$

ANSWER : (D) $80^{\circ}$

PROBLEM 25 :

The area of rectangle $A B C D$ is 72 . If point $A$ and the midpoints of $\overline{B C}$ and $\overline{C D}$ are joined to form a triangle, the area of that triangle is

(A) 21
(B) 27
(C) 30
(D) 36
(E) 40

ANSWER : (B) 27

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 :

AMERICAN MATHEMATICS COMPETITION - 2001

PROBLEM 1 :

Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?

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

ANSWER : (D) 10

PROBLEM 2 :

I'm thinking of two whole numbers. Their product is 24 and their sum is 11 . What is the larger number?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 12

ANSWER : (D) 8

PROBLEM 3 :

Granny Smith has $\$ 63$. Elberta has $\$ 2$ more than Anjou and Anjou has onethird as much as Granny Smith. How many dollars does Elberta have?
(A) 17
(B) 18
(C) 19
(D) 21
(E) 23

ANSWER : (E) 23

PROBLEM 4 :

The digits $1,2,3,4$ and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 9

ANSWER : (E) 9

PROBLEM 5 :

On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.
(A) 1
(B) $1 \frac{1}{2}$
(C) 2
(D) $2 \frac{1}{2}$
(E) 3

ANSWER : (C) 2

PROBLEM 6 :

Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?
(A) 90
(B) 100
(C) 105
(D) 120
(E) 140

ANSWER : (B) 100

Problems 7, 8 and 9 are about these kites.

PROBLEM 7 :

To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid.
What is the number of square inches in the area of the small kite?
(A) 21
(B) 22
(C) 23
(D) 24
(E) 25

ANSWER : (A) 21

PROBLEM 8 :

Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need?
(A) 30
(B) 32
(C) 35
(D) 38
(E) 39

ANSWER : (E) 39

PROBLEM 9 :

The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?
(A) 63
(B) 72
(C) 180
(D) 189
(E) 264

ANSWER : (D) 189

PROBLEM 10 :

A collector offers to buy state quarters for $2000 \%$ of their face value. At that rate how much will Bryden get for his four state quarters?
(A) 20 dollars
(B) 50 dollars
(C) 200 dollars
(D) 500 dollars
(E) 2000 dollars

ANSWER : (A) 20 dollars

PROBLEM 11 :

Points $A, B, C$ and $D$ have these coordinates: $A(3,2), B(3,-2), C(-3,-2)$ and $D(-3,0)$. The area of quadrilateral $A B C D$ is

(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

ANSWER : (C) 18

PROBLEM 12 :

If $a \otimes b=\frac{a+b}{a-b}$, then $(6 \otimes 4) \otimes 3==$
(A) 4
(B) 13
(C) 15
(D) 30
(E) 72

ANSWER : (A) 4

PROBLEM 13 :

Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
(A) 10
(B) 20
(C) 30
(D) 50
(E) 72

ANSWER : (D) 50

PROBLEM 14 :

Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?

Meat: beef, chicken, pork
Vegetables: baked beans, corn, potatoes, tomatoes
Dessert: brownies, chocolate cake, chocolate pudding, ice cream
(A) 4
(B) 24
(C) 72
(D) 80
(E) 144

ANSWER : (C) 72

PROBLEM 15 :

Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?
(A) 20
(B) 24
(C) 32
(D) 33
(E) 40

ANSWER : (A) 20

PROBLEM 16 :

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

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

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

PROBLEM 17 :

For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the following table (where $\mathrm{K}=1000$ ).

Between which two questions is the percent increase of the value the smallest?
(A) From 1 to 2v
(B) From 2 to 3
(C) From 3 to 4
(D) From 11 to 12
(E) From 14 to 15

ANSWER : (B) From 2 to 3

PROBLEM 18 :

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5 ?
(A) $\frac{1}{36}$
(B) $\frac{1}{18}$
(C) $\frac{1}{6}$
(D) $\frac{11}{36}$
(E) $\frac{1}{3}$

ANSWER : (D) $\frac{11}{36}$

PROBLEM 19 :

Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this?

ANSWER : (D)

PROBLEM 20 :

Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).
(A) $S, Q, M$
(B) Q,M,S
(C) Q,S,M
(D) $M, S, Q$
(E) $S, M, Q$

ANSWER : (A) $S, Q, M$

PROBLEM 21 :

The mean of a set of five different positive integers is 15 . The median is 18 . The maximum possible value of the largest of these five integers is
(A) 19
(B) 24
(C) 32
(D) 35
(E) 40

ANSWER : (D) 35

PROBLEM 22 :

On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?
(A) 90
(B) 91
(C) 92
(D) 95
(E) 97

ANSWER : (E) 97

PROBLEM 23 :

Points $R, S$ and $T$ are vertices of an equilateral triangle, and points $X, Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 20

ANSWER : (D) 4

PROBLEM 24 :

Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?

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

ANSWER : (B) 5

PROBLEM 25 :

There are 24 four-digit whole numbers that use each of the four digits 2,4 , 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
(A) 5724
(B) 7245
(C) 7254

(D) 7425

(E) 7542

ANSWER : (D) 7425

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.

American Mathematics Competition 8 - 2018

Question 1 :

An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?
(A) 14
(B) 15
(C) 16
(D) 18
(E) 20

Answer 1 :

(A) 14

Question 2 :

What is the value of the product

$$
\left(1+\frac{1}{1}\right) \cdot\left(1+\frac{1}{2}\right) \cdot\left(1+\frac{1}{3}\right) \cdot\left(1+\frac{1}{4}\right) \cdot\left(1+\frac{1}{5}\right) \cdot\left(1+\frac{1}{6}\right) ?
$$

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

Answer 2 :

(D) 7

Question 3 :

Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
(A) Arn
(B) Bob
(C) Cyd
(D) Dan
(E) Eve

Answer 3 :

(D) Dan

Question 4 :

The twelve-sided figure shown has been drawn on $1 \mathrm{~cm} \times 1 \mathrm{~cm}$ graph paper. What is the area of the figure in $\mathrm{cm}^2$ ?

(A) 12
(B) 12.5
(C) 13
(D) 13.5
(E) 14

Answer 4 :

Question 5 :

What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018 ?$
(A) -1010
(B) -1009
(C) 1008
(D) 1009
(E) 1010

Answer 5 :

(E) 1010

Question 6 :

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
(A) 50
(B) 70
(C) 80
(D) 90
(E) 100

Answer 6 :

Question 7 :

The 5 -digit number $\underline{2} \underline{0} \underline{1} \underline{8} \underline{U}$ is divisible by 9 . What is the remainder when this number is divided by 8 ?
(A) 1
(B) 3
(C) 5
(D) 6
(E) 7

Answer 7 :

(B) 3

Question 8 :

John Pork asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.

What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?
(A) 3.50
(B) 3.57
(C) 4.36
(D) 4.50
(E) 5.00

Answer 8 :

Question 9 :

Tyler is tiling the floor of his 12 -foot by 16 -foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?
(A) 48
(B) 87
(C) 89
(D) 96
(E) 120

Answer 9 :

(B) 87

Question 10 :

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1,2 , and 4 ?
(A) $\frac{3}{7}$
(B) $\frac{7}{12}$
(C) $\frac{12}{7}$
(D) $\frac{7}{4}$
(E) $\frac{7}{3}$

Answer 10 :

Question 11 :

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.

If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{7}{15}$
(D) $\frac{1}{2}$
(E) $\frac{2}{3}$

Answer 11 :

Question 12 :

The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
(A) $5: 50$
(B) $6: 00$
(C) $6: 30$
(D) $6: 55$
(E) $8: 10$

Answer 12 :

(B) $6: 00$

Question 13 :

John Pork took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100 , inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?
(A) 4
(B) 5
(C) 9
(D) 10
(E) 18

Answer 13 :

(A) 4

Question 14 :

Let $N$ be the greatest five-digit number whose digits have a product of 120 . What is the sum of the digits of $N$ ?
(A) 15
(B) 16
(C) 17
(D) 18
(E) 20

Answer 14 :

(D) 18

Question 15 :

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?

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

Answer 15 :

(D) 1

Question 16 :

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
(A) 1440
(B) 2880
(C) 5760
(D) 182,440
(E) 362,880

Answer 16 :

Question 17 :

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers $2 \frac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella?
(A) 704
(B) 845
(C) 1056
(D) 1760
(E) 3520

Answer 17 :

(A) 704

Question 18 :

How many positive factors does 23,232 have?
(A) 9
(B) 12
(C) 28
(D) 36
(E) 42

Answer 18 :

(E) 42

Question 19 :

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?

(A) 2
(B) 4
(C) 8
(D) 12
(E) 16

Answer 19 :

Question 20 :

In $\triangle A B C$, a point $E$ is on $\overline{A B}$ with $A E=1$ and $E B=2$. Point $D$ is on $\overline{A C}$ so that $\overline{D E} | \overline{B C}$ and point $F$ is on $\overline{B C}$ so that $\overline{E F} | \overline{A C}$. What is the ratio of the area of $C D E F$ to the area of $\triangle A B C ?

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

Answer 20 :

(A) $\frac{4}{9}$

Problem 21

How many positive three-digit integers have a remainder of 2 when divided by 6 , a remainder of 5 when divided by 9 , and a remainder of 7 when divided by 11 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer 21:

Problem 22

Point (E) is the midpoint of side (\overline{C D}) in square (A B C D), and (\overline{B E}) meets diagonal (\overline{A C}) at (F). The area of quadrilateral (A F E D) is 45 . What is the area of (A B C D) ?
(A) 100
(B) 108
(C) 120
(D) 135
(E) 144

Answer 22:

(B) 108

Problem 23

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

(A) 2/7
(B) 5/42
(C) 11/14
(D) 5/7
(E) 6/7

Answer 23:

(A) 2/7

Problem 24

In the cube (A B C D E F G H) with opposite vertices (C) and (E, J) and (I) are the midpoints of edges (\overline{F B}) and (\overline{H D}), respectively. Let (R) be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is (R^{2}) ?

Answer 24:

(D) Is the correct answer.

Problem 25

How many perfect cubes lie between (2^{8}+1) and (2^{18}+1), inclusive?
(A) 4
(B) 9
(C) 10
(D) 57
(E) 58

Answer 25:

(E) 58

American Mathematics Competition 8 - 2017

Question 1 :

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

Answer 1 :

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

Question 2 :

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

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

Answer 2 :

(E) 120

Question 3 :

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

Answer 3 :

Question 4 :

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

Answer 4 :

(D) 2400

Question 5 :

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

Answer 5 :

(B) 1120

Question 6 :

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

Answer 6 :

(D) 72

Question 7 :

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

Answer 7 :

(A) 11

Question 8 :

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7 .
(4) One of its digits is 9 .

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

Answer 8 :

(D) 8

Question 9 :

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

Answer 9 :

(D) 4

Question 10 :

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

Answer 10 :

Question 11 :

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

Answer 11 :

Question 12 :

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

Answer 12 :

(D) 60 and 79

Question 13 :

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

Answer 13 :

(B) 1

Question 14 :

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

Answer 14 :

Question 15 :

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

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

Answer 15 :

(D) 24

Question 16 :

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

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

Answer 16 :

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

Question 17 :

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

Answer 17 :

Question 18 :

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

(A) 12
(B) 24
(C) 26
(D) 30
(E) 36

Answer 18 :

(B) 24

Question 19 :

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

Answer 19 :

(D) 26

Question 20 :

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

Answer 20 :

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

Question 21 :

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

Answer 21 :

(A) 0

Question 22 :

In the right triangle $A B C, A C=12, B C=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

(A) $\frac{7}{6}$
(B) $\frac{13}{5}$
(C) $\frac{59}{18}$
(D) $\frac{10}{3}$
(E) $\frac{60}{13}$

Answer 22 :

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

Question 23 :

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

Answer 23 :

Question 24 :

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
(A) 78
(B) 80
(C) 144
(D) 146
(E) 152

Answer 24 :

(D) 146

Question 25 :

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

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

Answer 25 :

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

AMERICAN MATHEMATICS COMPETITION 8 - 2025

PROBLEM 1 :

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4 \times 4$ grid is covered by the star?

(A) 40
(B) 50
(C) 60
(D) 75
(E) 80

ANSWER :

(B) 50

PROBLEM 2 :

The table below shows the ancient Egyptian hieroglyphs that were used to represent different numbers.

For example, the number 32 was represented by the hieroglyphs $\cap \cap \cap |$. What number is represented by the following combination of hieroglyphs?

(A) 1,423
(B) 10,423
(C) 14,023
(D) 14,203
(E) 14,230

ANSWER :

(B) 10,423

PROBLEM 3 :

Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and 3 of her friends play Buffalo Shuffle-o, each player is dealt 15 cards. Suppose 2 more friends join the next game. How many cards will be dealt to each player?

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

ANSWER :

PROBLEM 4 :

Lucius is counting backward by 7 s . His first three numbers are 100,93 , and 86 . What is his 10 th number?
(A) 30
(B) 37
(C) 42
(D) 44
(E) 47

ANSWER :

(B) 37

PROBLEM 5 :

Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labled $F$ ) and drives to location $A$, then $B$, then $C$, before returning to $F$. What is the shortest distance, in blocks, she can drive to complete the route?

(A) 20
(B) 23
(C) 24
(D) 26
(E) 28

ANSWER :

PROBLEM 6 :

Sekou writes the numbers $15,16,17,18,19$. After he erases one of his numbers, the sum of the remaining four numbers is a multiple of 4 . Which number did he erase?
(A) 15
(B) 16
(C) 17
(D) 18
(E) 19

ANSWER :

PROBLEM 7 :

On the most recent exam on Prof. Xochi's class,

5 students earned a score of at least $95 \%$,
13 students earned a score of at least $90 \%$,
27 students earned a score of at least $85 \%$,
50 students earned a score of at least $80 \%$,

How many students earned a score of at least $80 \%$ and less than $90 \%$ ?
(A) 8
(B) 14
(C) 22
(D) 37
(E) 45

ANSWER :

(D) 37

PROBLEM 8 :

Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of 18 square centimeters. What is the volume of the cube in cubic centimeters?

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

ANSWER :

(A) $3 \sqrt{3}$

PROBLEM 9 :

Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?

(A) 5
(B) 6.5
(C) 8
(D) 9.5
(E) 12

ANSWER :

(B) 6.5

PROBLEM 10 :

In the figure below, $A B C D$ is a rectangle with sides of length $A B=5$ inches and $A D=3$ inches. Rectangle $A B C D$ is rotated $90^{\circ}$ clockwise around the midpoint of side $D C$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?

(A) 21
(B) 22.25
(C) 23
(D) 23.75
(E) 25

ANSWER :

(D) 23.75

PROBLEM 11 :

A tetromino consists of four squares connected along their edges. There are five possible tetromino shapes, $I, O, L, T$, and $S$, shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a $3 \times 4$ rectangle. At least one of the tiles is an $S$ tile. What are the other two tiles?

(A) $I$ and $L$
(B) $I$ and $T$
(C) $L$ and $L$
(D) $L$ and $S$
(E) $O$ and $T$

ANSWER :

PROBLEM 12 :

The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?

(A) $3 \pi$
(B) $4 \pi$
(C) $5 \pi$
(D) $6 \pi$
(E) $8 \pi$

ANSWER :

PROBLEM 13 :

Each of the even numbers $2,4,6, \ldots, 50$ is divided by 7 . The remainders are recorded. Which histogram displays the number of times each remainder occurs?

ANSWER :

(A)

PROBLEM 14 :

A number $N$ is inserted into the list $2,6,7,7,28$. The mean is now twice as great as the median. What is $N$ ?
(A) 7
(B) 14
(C) 20
(D) 28
(E) 34

ANSWER :

(E) 34

PROBLEM 15 :

Kei draws a 6 -by- 6 grid. He colors 13 of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of $m+M$ ?

(A) 12
(B) 14
(C) 16
(D) 18
(E) 20

ANSWER :

PROBLEM 16 :

Five distinct integers from 1 to 10 are chosen, and five distinct integers from 11 to 20 are chosen. No two numbers differ by exactly 10 . What is the sum of the ten chosen numbers?
(A) 95
(B) 100
(C) 105
(D) 110
(E) 115

ANSWER :

PROBLEM 17 :

In the land of Markovia, there are three cities: $A, B$, and $C$. There are 100 people who live in $A, 120$ who live in $B$, and 160 who live in $C$. Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, $\frac{1}{4}$ of the people who live in $A$ work in $B$.) How many people work in $A$ ?

(A) 55
(B) 60
(C) 85
(D) 115
(E) 160

ANSWER :

(D) 115

PROBLEM 18 :

The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R$, in units, of the circle on the right?

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

ANSWER :

(B) 2

PROBLEM 19 :

Two towns, $A$ and $B$, are connected by a straight road that is 15 miles long. Travelling from city $A$ to town $B$, the speed limit changes every 5 miles: from 25 to 40 to 20 miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet?

(A) 7.75
(B) 8
(C) 8.25
(D) 8.5
(E) 8.75

ANSWER :

(D) 8.5

PROBLEM 20 :

Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
(A) $\frac{4}{7}$
(B) $\frac{3}{5}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$
(E) $\frac{7}{8}$

ANSWER :

(A) $\frac{4}{7}$.

PROBLEM 21 :

The Konigsberg School has assigned grades 1 through 7 to pods $A$ through $G$, one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by 2 or more grade levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E$, and $F$ ?

(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

ANSWER :

(A) 12

PROBLEM 22 :

A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina's pattern?

(A) 2
(B) 4
(C) 5
(D) 7
(E) 9

ANSWER :

(D) 7

PROBLEM 23 :

How many four-digit numbers have all three of the following properties?
(I) The tens and ones digit are both 9 .
(II) The number is 1 less than a perfect square.
(III) The number is the product of exactly two prime numbers.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

(B) 1

PROBLEM 24 :

In trapezoid $A B C D$, angles $B$ and $C$ measure $60^{\circ}$ and $A B=D C$. The side lengths are all positive integers, and the perimeter of $A B C D$ is 30 units. How many non-congruent trapezoids satisfy all of these conditions?

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

ANSWER :

(E) 4

PROBLEM 25 :

Makayla finds all the possible ways to draw a path in a $5 \times 5$ diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?

(A) 2520
(B) 3150
(C) 3840
(D) 4730
(E) 5050

ANSWER :

(B) 3150