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 - 2003

PROBLEM 1 :

Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum?

(A) 12
(B) 16
(C) 20
(D) 22
(E) 26

ANSWER : (E) 26

PROBLEM 2 :

Which of the following numbers has the smallest prime factor?
(A) 55
(B) 57
(C) 58
(D) 59
(E) 61

ANSWER : (C) 58

PROBLEM 3 :

A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?
(A) $60 \%$
(B) $65 \%$
(C) $70 \%$
(D) $75 \%$
(E) $90 \%$

ANSWER : (D) $75 \%$

PROBLEM 4 :

A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted 7 children and 19 wheels. How many tricycles were there?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 7

ANSWER : (C) 5

PROBLEM 5 :

If $20 \%$ of a number is 12 , what is $30 \%$ of the same number?
(A) 15
(B) 18
(C) 20
(D) 24
(E) 30

ANSWER : (B) 18

PROBLEM 6 :

Given the areas of the three squares in the figure, what is the area of the interior triangle?

(A) 13
(B) 30
(C) 60
(D) 300
(E) 1800

ANSWER : (B) 30

PROBLEM 7 :

Blake and Jenny each took four 100 -point tests. Blake averaged 78 on the four tests. Jenny scored 10 points higher than Blake on the first test, 10 points lower than him on the second test, and 20 points higher on both the third and fourth tests. What is the difference between Jenny's average and Blake's average on these four tests?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 40

ANSWER : (A) 10

Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures.

Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

Art's cookies are trapezoids:

Roger's cookies are rectangles:

Paul's cookies are parallelograms:

Trisha's cookies are triangles:

Each friend uses the same amount of dough, and Art makes exactly 12 cookies.

PROBLEM 8 :

Who gets the fewest cookies from one batch of cookie dough?
(A) Art

(B) Paul

(D) Trisha

(E) There is a tie for fewest.

ANSWER : (A) Art

PROBLEM 9 :

Art's cookies sell for $60 ¢$ each. To earn the same amount from a single batch, how much should one of Roger's cookies cost?
(A) $18 ¢$
(B) $25 ¢$
(C) $40 ¢$
(D) $75 ¢$
(E) $90 ¢$

ANSWER : (C) $40 ¢$

PROBLEM 10 :

How many cookies will be in one batch of Trisha's cookies?
(A) 10
(B) 12
(C) 16
(D) 18
(E) 24

ANSWER : (E) 24

PROBLEM 11 :

Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10 \%$. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $\$ 40$ on Thursday?
(A) $\$ 36$
(B) $\$ 39.60$
(C) $\$ 40$
(D) $\$ 40.40$
(E) $\$ 44$

ANSWER : (B) $\$ 39.60$

PROBLEM 12 :

When a fair six-sided die is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the numbers on the five faces that can be seen is divisible by 6 ?
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{5}{6}$
(E) 1

ANSWER : (E) 1

PROBLEM 13 :

Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?

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

ANSWER : (B) 6

PROBLEM 14 :

In this addition problem, each letter stands for a different digit.

If $T=7$ and the letter $O$ represents an even number, what is the only possible value for $W$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER : (D) 3

PROBLEM 15 :

A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?

(A) 3

(B) 4

(D) 6

(E) 7

ANSWER : (B) 4

PROBLEM 16 :

Ali, Bonnie, Carlo and Dianna are going to drive together to a nearby theme park. The car they are using has four seats: one driver's seat, one front passenger seat and two back seats. Bonnie and Carlo are the only two who can drive the car. How many possible seating arrangements are there?
(A) 2
(B) 4
(C) 6
(D) 12
(E) 24

ANSWER : (D) 12

PROBLEM 17 :

The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?

(A) Nadeen and Austin
(B) Benjamin and Sue
(C) Benjamin and Austin
(D) Nadeen and Tevyn
(E) Austin and Sue

ANSWER : (E) Austin and Sue

PROBLEM 18 :

Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?

(A) 1
(B) 4
(C) 5
(D) 6
(E) 7

ANSWER : (D) 6

PROBLEM 19 :

How many integers between 1000 and 2000 have all three of the numbers 15,20 and 25 as factors?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

ANSWER : (C) 3

PROBLEM 20 :

What is the measure of the acute angle formed by the hands of a clock at 4:20 a.m.?
(A) $0^{\circ}$
(B) $5^{\circ}$
(C) $8^{\circ}$
(D) $10^{\circ}$
(E) $12^{\circ}$

ANSWER : (D) $10^{\circ}$

PROBLEM 21 :

The area of trapezoid $A B C D$ is $164 \mathrm{~cm}^{2}$. The altitude is $8 \mathrm{~cm}, A B$ is 10 cm , and $C D$ is 17 cm . What is $B C$, in centimeters?

(A) 9
(B) 10
(C) 12
(D) 15
(E) 20

ANSWER : (B) 10

PROBLEM 22 :

The following figures are composed of squares and circles. Which figure has a shaded region with largest area?

(A) A only

(B) B only

(D) both A and B

(E) all are equal

ANSWER : (C) C only

PROBLEM 23 :

In the pattern below, the cat moves clockwise through the four squares and the mouse moves counterclockwise through the eight exterior segments of the four squares.

If the pattern is continued, where would the cat and mouse be after the 247th move?

ANSWER : (A)

PROBLEM 24 :

A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?

ANSWER : (B)

PROBLEM 25 :

In the figure, the area of square $W X Y Z$ is $25 \mathrm{~cm}^{2}$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\triangle A B C, A B=A C$, and when $\triangle A B C$ is folded over side $\overline{B C}$, point $A$ coincides with $O$, the center of square $W X Y Z$. What is the area of $\triangle A B C$, in square centimeters?

(A) $\frac{15}{4}$
(B) $\frac{21}{4}$
(C) $\frac{27}{4}$
(D) $\frac{21}{2}$
(E) $\frac{27}{2}$

ANSWER : (C) $\frac{27}{4}$

AMERICAN MATHEMATICS COMPETITION 8 - 2002

PROBLEM 1 :

A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

ANSWER : (D) 5

PROBLEM 2 :

How many different combinations of $\$ 5$ bills and $\$ 2$ bills can be used to make a total of $\$ 17$ ? Order does not matter in this problem.

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

ANSWER : (A) 2

PROBLEM 3 :

What is the smallest possible average of four distinct positive even integers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

ANSWER : (C) 5

PROBLEM 4 :

The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?
(A) 0
(B) 4
(C) 9
(D) 16
(E) 25

ANSWER : (B) 4

PROBLEM 5 :

Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old?
(A) Monday
(B) Wednesday
(C) Friday
(D) Saturday (E) Sunday

ANSWER : (C) Friday

PROBLEM 6 :

A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it?

(A) A
(B) B
(C) C
(D) D
(E) E

ANSWER : (A) A

PROBLEM 7 :

The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?

(A) 5
(B) 12
(C) 15
(D) 16
(E) 20

ANSWER : (E) 20

Problems 8,9 and 10 use the data found in the accompanying paragraph and table:

Juan's Old Stamping Grounds :

Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6 ¢$ each, Peru $4 \phi$ each, and Spain 5¢ each. (Brazil and Peru are South American countries and France and Spain are in Europe.)

PROBLEM 8 :

How many of his European stamps were issued in the '80s?\
(A) 9
(B) 15
(C) 18
(D) 24
(E) 42

ANSWER : (D) 24

PROBLEM 9 :

His South American stamps issued before the '70s cost him
(A) $\$ 0.40$
(B) $\$ 1.06$
(C) $\$ 1.80$
(D) $\$ 2.38$
(E) $\$ 2.64$

ANSWER : (B) $\$ 1.06$

PROBLEM 10 :

The average price of his '70s stamps is closest to
(A) $3.5 ¢$
(B) $4 ¢$
(C) $4.5 ¢$
(D) $5 ¢$
(E) $5.5 ¢$

ANSWER : (E) $5.5 ¢$

PROBLEM 11 :

A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?

(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

ANSWER : (C) 13

PROBLEM 12 :

A board game spinner is divided into three regions labeled $A, B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. The probability of the arrow stopping on region $C$ is

(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{5}$
(D) $\frac{1}{3}$
(E) $\frac{2}{5}$

ANSWER : (B) $\frac{1}{6}$

PROBLEM 13 :

For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert's. Approximately, how many jellybeans did Carrie get?
(A) 250
(B) 500
(C) 625
(D) 750
(E) 1000

ANSWER : (E) 1000

PROBLEM 14 :

A merchant offers a large group of items at $30 \%$ off. Later, the merchant takes $20 \%$ off these sale prices and claims that the final price of these items is $50 \%$ off the original price. The total discount is
(A) $35 \%$
(B) $44 \%$
(C) $50 \%$
(D) $56 \%$
(E) $60 \%$

ANSWER : (B) $44 \%$

PROBLEM 15 :

Which of the following polygons has the largest area?

(A) A
(B) B
(C) C
(D) D
(E) E

ANSWER : (E) E

PROBLEM 16 :

Right isosceles triangles are constructed on the sides of a $3-4-5$ right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?

(A) $X+Z=W+Y$
(B) $W+X=Z$
(C) $3 X+4 Y=5 Z$
(D) $X+W=\frac{1}{2}(Y+Z)$
(E) $X+Y=Z$

ANSWER : (E) $X+Y=Z$

PROBLEM 17 :

In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29 , how many correct answers did she have?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

ANSWER : (C) 7

PROBLEM 18 :

Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
(A) 1 hr

(B) 1 hr 10 min

(D) 1 hr 40 min

(E) 2 hr

ANSWER : (E) 2 hr

PROBLEM 19 :

How many whole numbers between 99 and 999 contain exactly one 0 ?
(A) 72
(B) 90
(C) 144
(D) 162
(E) 180

ANSWER : (D) 162

PROBLEM 20 :

The area of triangle $X Y Z$ is 8 square inches. Points $A$ and $B$ are midpoints of congruent segments $\overline{X Y}$ and $\overline{X Z}$. Altitude $\overline{X C}$ bisects $\overline{Y Z}$. The area (in square inches) of the shaded region is

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

ANSWER : (D) 3

PROBLEM 21 :

Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is
(A) $\frac{5}{16}$
(B) $\frac{3}{8}$
(C) $\frac{1}{2}$
(D) $\frac{5}{8}$
(E) $\frac{11}{16}$

ANSWER : (E) $\frac{11}{16}$

PROBLEM 22 :

Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.

(A) 18
(B) 24
(C) 26
(D) 30
(E) 36

ANSWER : (C) 26

PROBLEM 23 :

A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?

(A) $\frac{1}{3}$
(B) $\frac{4}{9}$
(C) $\frac{1}{2}$
(D) $\frac{5}{9}$
(E) $\frac{5}{8}$

ANSWER : (B) $\frac{4}{9}$

PROBLEM 24 :

Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

ANSWER : (B) 40

PROBLEM 25 : Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?\
(A) $\frac{1}{10}$
(B) $\frac{1}{4}$
(C) $\frac{1}{3}$
(D) $\frac{2}{5}$
(E) $\frac{1}{2}$

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

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 - 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: