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:

(C) $\frac{1}{8}$

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:

(C) 139.

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:

(C) 4.

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:

(C) 87431.

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:

(C) 9.

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:

(C) 6.

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:

(C) $\frac{1}{2}$

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.

(C) The gardens are the same size.

(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:

(C) Is the correct 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:

(C) Is the correct 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:

(C) 3.3

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:

(C) 20

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:

(C) Is the correct 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:

(C) Is the corret 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

(C) 10

(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:

(C) 37

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 :

(C) 13

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 :

(C) 80

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 :

(C) 4.36

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 :

(C) $\frac{12}{7}$

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 :

(C) $\frac{7}{15}$

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 :

(C) 5760

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 :

(C) 8

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:

(C) 3

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 :

(C) 8

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 :

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

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 :

(C) 361

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 :

(C) 93

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 :

(C) 45

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 :

(C) 25

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 - 2013

Problem 1

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars.

What is the smallest number of additional cars she must buy in order to be able to arrange her cars in this way?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer :

(A) 1

Problem 2

A sign at the fish market says, " $50 \%$ off, today only: half-pound packages for just $\$ 3$ per package."

What is the regular price for a full pound of fish, in dollars?
(A) 6
(B) 9
(C) 10
(D) 12
(E) 15

Answer:

(D) 12

Problem 3


What is the value of $4 \cdot(-1+2-3+4-5+6-7+\cdots+1000)$ ?
(A) -10
(B) 0
(C) 1
(D) 500
(E) 2000

Answer:

(E) 2000

Problem 4

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money,

each of her seven friends paid an extra $\$ 2.50$ to cover her portion of the total bill. What was the total bill?
(A) $\$ 120
(B) $\$ 128
(C) $\$ 140
(D) $\$ 144
(E) $\$ 160

Answer:

(C) $\$ 140

Problem 5

Hammie is in the $6^{\text {th }}$ grade and weighs 106 pounds.

His quadruplet sisters are tiny babies and weigh $5,5,6$, and 8 pounds.

Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
(A) median, by 60
(B) median, by 20
(C) average, by 5
(D) average, by 15
(E) average, by 20

Answer:

(E) average, by 20

Problem 6


The number in each box below is the product of the numbers in the two boxes that touch it in the row above.

For example, $30=6 \times 5$. What is the missing number in the top row?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Answer:

(C) 4

Peoblem 7


Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass,

Trey counted 6 cars in the first 10 seconds.

It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed.

Which of the following was the most likely number of cars in the train?
(A) 60
(B) 80
(C) 100
(D) 120
(E) 140

Answer:

(C) 100

Problem 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
(A) 1/8
(B) 1/4
(C) 3/8
(D) 1/2
(E) 3/4

Answer:

(C) 3/8

Problem 9
The Incredible Hulk can double the distance he jumps with each succeeding jump.

If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on,

then on which jump will he first be able to jump more than 1 kilometer?
(A) 9th
(B) 10th
(C) 11th
(D) 12th
(E) 13th

Answer:

(C) 11th

Problem 10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
(A) 110
(B) 165
(C) 330
(D) 625
(E) 660

Answer:

(C) 330

Problem 11
Ted's grandfather used his treadmill on 3 days this week.

He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour.

He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday.

If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill.

How many minutes less?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer:

(D) 4

Problem 12
At the 2013 Winnebago Country Fair a vendor is offering a "fair special" on sandals.

If you buy one pair of sandals at the regular price of $\$ 50$, you get a second pair at a $40 \%$ discount,

and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals.

What percentage of the $\$ 150$ regular price did he save?
(A) 25
(B) 30
(C) 33
(D) 40
(E) 45

Answer:

(B) 30

Problem 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score.

By which of the following might her incorrect sum have differed from the correct one?
(A) 45
(B) 46
(C) 47
(D) 48
(E) 49

Answer:

(A) 45

Problem 14
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand.

Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
(A) 1/4
(B) 1/3
(C) 3/8
(D) 1/2
(E) 2/3

Answer:

(C) 3/8

Problem 15
If $3^{p}+3^{4}=90,2^{r}+44=76$, and $5^{3}+6^{s}=1421$, what is the product of $p, r$, and $s$ ?
(A) 27
(B) 40
(C) 50
(D) 70
(E) 90

Answer:

(B) 40

Problem 16
A number of students from Fibonacci Middle School are taking part in a community service project.

The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $5: 3$, and the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$-graders is $8: 5$.

What is the smallest number of students that could be participating in the project?
(A) 16
(B) 40
(C) 55
(D) 79
(E) 89

Answer:

(E) 89

Problem 17
The sum of six consecutive positive integers is 2013 . What is the largest of these six integers?
(A) 335
(B) 338
(C) 340
(D) 345
(E) 350

Answer:

(B) 338

Problem 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high.

The floor and the four walls are all one foot thick. How many blocks does the fort contain?
(A) 204
(B) 280
(C) 320
(D) 340
(E) 600

Answer:

(B) 280

Problem 19
Bridget, Cassie, and Hannah are discussing the results of their last math test.

Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show their tests to anyone.

Cassie says, "I didn't get the lowest score in our class," and Bridget adds, "I didn't get the highest score.

" What is the ranking of the three girls from highest to lowest?
(A) Hannah, Cassie, Bridget
(B) Hannah, Bridget, Cassie
(C) Cassie, Bridget, Hannah
(D) Cassie, Hannah, Bridget
(E) Bridget, Cassie, Hannah

Answer:

(D) Cassie, Hannah, Bridget

Problem 20
A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter.

What is the area of the semicircle?
(A) $\frac{\pi}{2}$
(B) $\frac{2 \pi}{3}$
(C) $\pi$
(D) $\frac{4 \pi}{3}$
(E) $\frac{5 \pi}{3}$

Answer:

(C) $\pi$

Problem 21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park.

Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park.

On school days she bikes on streets to the southwest corner of City Park,

then takes a diagonal path through the park to the northeast corner of City Park, and then bikes on streets to school.

If her route is as short as possible, how many different routes can she take?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 18

Answer:

(E) 18

Problem 22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks high.

How many toothpicks are used altogether?
(A) 1920
(B) 1952
(C) 1980
(D) 2013
(E) 3932

Answer:

(E) 3932

Problem 23
Angle $A B C$ of $\triangle A B C$ is a right angle.

The sides of $\triangle A B C$ are the diameters of semicircles as shown.

The area of the semicircle on $\overline{A B}$ equals $8 \pi$,

and the arc of the semicircle on $\overline{A C}$ has length $8.5 \pi$.

What is the radius of the semicircle on $\overline{B C}$ ?
(A) 7
(B) 7.5
(C) 8
(D) 8.5
(E) 9

Answer:

(B) 7.5

Problem 24
Squares $A B C D, E F G H$, and $G H I J$ are equal in area. Points $C$ and $D$ are the midpoints of sides $I H$ and $H E$, respectively. What is the ratio of the area of the shaded pentagon $A J I C B$ to the sum of the areas of the three squares?


(A) $\frac{1}{4}$
(B) $\frac{7}{24}$
(C) $\frac{1}{3}$
(D) $\frac{3}{8}$
(E) $\frac{5}{12}$

Answer:

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

Problem 25
A ball with diameter 4 inches starts at point $A$ to roll along the track shown.

The track is comprised of 3 semicircular arcs whose radii are $R_{1}=100$ inches,

$R_{2}=60$ inches, and $R_{3}=80$ inches, respectively.

The ball always remains in contact with the track and does not slip.

What is the distance in inches the center of the ball travels over the course from $A$ to $B$ ?
(A) $238 \pi$
(B) $240 \pi$
(C) $260 \pi$
(D) $280 \pi$
(E) $500 \pi$

Answer:

(A) $238 \pi$

AMERICAN MATHEMATICS COMPETITION 8 - 2014

Problem 1

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$ ?
(A) -10
(B) -6
(C) 0
(D) 6
(E) 10

Answer

(A) -10

Problem 2

Paul owes Paula 35 cents and has a pocket full of 5 -cent coins, 10 -cent coins, and 25 -cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer

(E) 5

Problem 3

Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
(A) 240
(B) 250
(C) 260
(D) 270
(E) 280

Answer

(B) 250 pages.

Problem 4

The sum of two prime numbers is 85 . What is the product of these two prime numbers?
(A) 85
(B) 91
(C) 115
(D) 133
(E) 166

Answer

(E) 166.

Problem 5

Margie's car can go 32 miles on a gallon of gas, and gas currently costs $\$ 4$ per gallon. How many miles can Margie drive on $\$ 20$ worth of gas?
(A) 64
(B) 128
(C) 160
(D) 320
(E) 640

Answer

(C) 160.

Problem 6

Six rectangles each with a common base width of 2 have lengths of $1,4,9,16,25$, and 36 . What is the sum of the areas of the six rectangles?
(A) 91
(B) 93
(C) 162
(D) 182
(E) 202

Answer

(D) 182.

Problem 7

There are four more girls than boys in Ms. Raub's class of 28 students. What is the ratio of number of girls to the number of boys in her class?
(A) $3: 4$
(B) $4: 3$
(C) $3: 2$
(D) $7: 4$
(E) $2: 1$

Answer

(B) $4: 3$

Problem 8

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\$ \underline{1 A 2}$. What is the missing digit $A$ of this 3 -digit number?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer

(D) 3 $\sim$ fn106068.

Problem 9

Answer

(D) 140.

Problem 10

The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8 . In what year was Samantha born?
(A) 1979
(B) 1980
(C) 1981
(D) 1982
(E) 1983

Answer

(A) $1979 \sim$ SweetMango77
corrections made by DrDominic.

Problem 11

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
(A) 4
(B) 5
(C) 6
(D) 8
(E) 10

Answer

(A) 4 is the correct answer.

Problem 12

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?

Answer

(B) Is the correct answer.

Problem 13

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

$\textbf{(A) }$ $n$ and $m$ are even $\qquad\textbf{(B) }$ $n$ and $m$ are odd $\qquad\textbf{(C) }$ $n+m$ is even $\qquad\textbf{(D) }$ $n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

Answer

(D) is odd.

Problem 14

Rectangle 

$ABCD$

 and right triangle 

$DCE$

 have the same area. They are joined to form a trapezoid, as shown. What is 

$DE$

?

Answer

(B) Is the correct answer.

Problem 15

The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$ ?

(A) 75

(B) 80

(C) 90

(D) 120

(E) 150

Answer

(C) 90.

Problem 16

The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against nonconference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?

(A) 60

(B) 88

(C) 96

(D) 144

(E) 160

Answer

(B) 88.

Problem 17

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?

(A) 4

(B) 6

(C) 8

(D) 10

(E) 12

Answer

(B) 6.

Problem 18

Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?

$(\mathbf{A})$ all 4 are boys $(\mathbf{B})$ all 4 are girls $(\mathbf{C})_{2}$ are girls and 2 are boys $(\mathbf{D})_{3}$ are of one gender and 1 is of the other gender $\boldsymbol{(} \mathbf{E} \boldsymbol{)}$ all of these outcomes are equally likely

Answer

(D) Is the correct answer.

Problem 19

A cube with 3 -inch edges is to be constructed from 27 smaller cubes with 1 -inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3 -inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

(A) $\frac{5}{54}$

(B) $\frac{1}{9}$

(C) $\frac{5}{27}$

(D) $\frac{2}{9}$

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

Answer

(A) $\frac{5}{54}$

Problem 20

Rectangle ABCD has sides $\mathrm{CD}=3$ and $\mathrm{DA}=5$. A circle of radius 1 is centered at A , a circle of radius 2 is centered at B , and a circle of radius 3 is centered at C . Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

(A) 3.5

(B) 4.0

(C) 4.5

(D) 5.0

(E) 5.5

Answer

(B) 4.0

Problem 21

The 7-digit numbers $\underline{74 A 52 B 1}$ and $\underline{326 A B 4 C}$ are each multiples of 3 . Which of the following could be the value of $C$ ?

(A) 1

(B) 2

(C) 3

(D) 5

(E) 8

Answer

(A) 1.

Problem 22

A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?

(A) 1

(B) 3

(C) 5

(D) 7

(E) 9

Answer

(E) 9.

Problem 23

Three members of the Euclid Middle School girls' softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all 2 -digit primes.

Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today's date.

What number does Caitlin wear?

(A) 11

(B) 13

(C) 17

(D) 19

(E) 23

Answer

(A) 11.

Problem 24

One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

(A) 2.5

(B) 3.0

(C) 3.5

(D) 4.0

(E) 4.5

Answer

(C) 3.5

Problem 25

A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?

Note: 1 mile= 5280 feet

(A) $\frac{\pi}{11}$

(B) $\frac{\pi}{10}$

(C) $\frac{\pi}{5}$

(D) $\frac{2 \pi}{5}$

(E) $\frac{2 \pi}{3}$

Answer

(B) $\frac{\pi}{10}$

American Mathematics Competition 8 - 2009

Problem 1

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
(A) 3
(B) 4
(C) 7
(D) 11
(E) 14

Answer: The answer is (E) 14

Problem 2

On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
(A) 7
(B) 32
(C) 35
(D) 49
(E) 112

Answer: $($ D $) 49$

Problem 3

The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7

Answer : (C) 6

Problem 4

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?

Answer : The answer is (B)

Problem 5

A sequence of numbers starts with 1,2 , and 3 . The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
(A) 11
(B) 20
(C) 37
(D) 68
(E) 99

Answer : D

Problem 6

Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?
(A) 40
(B) 42
(C) 44
(D) 46
(E) 48

Answer : A

Problem 7

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?

(A) 2
(B) 3
(C) 4.5
(D) 6
(E) 9

Answer : C

Problem 8

The length of a rectangle is increased by $10 \%$ and the width is decreased by $10 \%$. What percent of the old area is the new area?
(A) 90
(B) 99
(C) 100
(D) 101
(E) 110

Answer : B

Problem 9

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

Answer : B

Problem 10

On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?

Answer : D

Problem 11

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $\$ 1.43$. Some of the 30 sixth graders each bought a pencil, and they paid a total of $\$ 1.95$. How many more sixth graders than seventh graders bought a pencil?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Answer : D

Problem 12

The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?

(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{7}{9}$
(E) $\frac{5}{6}$

Answer : D

Problem 13

A three-digit integer contains one of each of the digits 1,3 , and 5 . What is the probability that the integer is divisible by 5 ?
(A) $\frac{1}{6}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$
(E) $\frac{5}{6}$

Answer : B

Problem 14

Austin and Temple are 50 miles apart along Interstate 35 . Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
(A) 46
(B) 48
(C) 50
(D) 52
(E) 54

Answer : B

Problem 15

A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, $1 / 4$ cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?
(A) $5 \frac{1}{8}$
(B) $6 \frac{1}{4}$
(C) $7 \frac{1}{2}$
(D) $8 \frac{3}{4}$
(E) $9 \frac{7}{8}$

Answer : D

Problem 16

How many 3 -digit positive integers have digits whose product equals 24 ?
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

Answer : D

Problem 17

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of 360 and $x$ is a square and the product of 360 and $y$ is a cube. What is the sum of $x$ and $y$ ?
(A) 80
(B) 85
(C) 115
(D) 165
(E) 610

Answer : B

Problem 18

The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed?

(A) 49
(B) 57
(C) 64
(D) 96
(E) 126

Answer : C

Problem 19

Two angles of an isosceles triangle measure $70^{\circ}$ and $x^{\circ}$. What is the sum of the three possible values of $x$ ?
(A) 95
(B) 125
(C) 140
(D) 165
(E) 180

Answer : D

Problem 20

How many non-congruent triangles have vertices at three of the eight points in the array shown below?

Answer : D

Problem 21

Andy and Bethany have a rectangular array of numbers greater than 0 with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is $A$. Bethany adds the numbers in each column. The average of her 75 sums is $B$. What is the value of $\frac{A}{B}$ ?
(A) $\frac{64}{225}$
(B) $\frac{8}{15}$
(C) 1
(D) $\frac{15}{8}$
(E) $\frac{225}{64}$

Answer : D

Problem 22

How many whole numbers between 1 and 1000 do not contain the digit 1 ?
(A) 512
(B) 648
(C) 720
(D) 728
(E) 800

Answer : D

Problem 23

On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
(A) 26
(B) 28
(C) 30
(D) 32
(E) 34

Answer : B

Problem 24

The letters $A, B, C$ and $D$ represent digits. If $AB+CA=DA$ and $AB-C A=A $, what digit does $D$ represent?

Answer : E

Problem 25

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $1 / 2$ foot from the top face. The second cut is $1 / 3$ foot below the first cut, and the third cut is $1 / 17$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?

(A) 6
(B) 7
(C) $\frac{419}{51}$
(D) $\frac{158}{17}$
(E) 11

Answer : E

AMERICAN MATHEMATICS COMPETITION 8 - 2023

PROBLEM 1 :

What is the value of $(8 \times 4+2)-(8+4 \times 2)$ ?
(A) 0
(B) 6
(C) 10
(D) 18
(E) 24

ANSWER :

(D) 18

PROBLEM 2 :

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?

ANSWER :

(E)

PROBLEM 3 :

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation

$$
(\text { wind chill })=(\text { air temperature })-0.7 \times(\text { wind speed }),
$$

where temperature is measured in degrees Fahrenheit ( ${ }^{\circ} \mathrm{F}$ ) and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ} \mathrm{F}$ and the wind speed is 18 mph . Which of the following is closest to the approximate wind chill?
(A) 18
(B) 23
(C) 28
(D) 32
(E) 35

ANSWER :

(B) 23

PROBLEM 4 :

The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7 . How many of these four numbers are prime?

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

ANSWER :

(D) 3

PROBLEM 5 :

A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
(A) 1250
(B) 1500
(C) 1750
(D) 1800
(E) 2000

ANSWER :

(B) 1500

PROBLEM 6 :

The digits $2,0,2$, and 3 are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

(A) 0
(B) 8
(C) 9
(D) 16
(E) 18

ANSWER :

(C) 9

PROBLEM 7 :

A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15,3)$ and $(16,5)$. A line is drawn through points $A(0,0)$ and $B(3,1)$. Another line is drawn through points $C(0,10)$ and $D(2,9)$. How many points on the rectangle lie on at least one of the two lines?

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

ANSWER :

(B) 1

PROBLEM 8 :

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo's win-loss record?

(A) 000101
(B) 001001
(C) 010000
(D) 010101
(E) 011000

SOLUTION :

(A) 000101

PROBLEM 9 :

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between 4 and 7 meters?

(A) 6
(B) 8
(C) 10
(D) 12
(E) 14

ANSWER :

(B) 8

PROBLEM 10 :

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?
(A) $\frac{1}{12}$
(B) $\frac{1}{6}$
(C) $\frac{1}{4}$
(D) $\frac{1}{3}$
(E) $\frac{5}{12}$

ANSWER :

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

PROBLEM 11 :

NASA's Perseverance Rover was launched on July 30 , 2020. After traveling $292,526,838$ miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
(A) 6,000
(B) 12,000
(C) 60,000
(D) 120,000
(E) 600,000

ANSWER :

(C) 60,000

PROBLEM 12 :

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

(A) $\frac{1}{4}$
(B) $\frac{11}{36}$
(C) $\frac{1}{3}$
(D) $\frac{19}{36}$
(E) $\frac{5}{9}$

ANSWER :

(B) $\frac{11}{36}$

PROBLEM 13 :

Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the start and finish lines. The 3 rd water station is located 2 miles after the 1 st repair station. How long is the race in miles?

(A) 8
(B) 16
(C) 24
(D) 48
(E) 96

ANSWER :

(D) 48

PROBLEM 14 :

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5 -cent, 10 -cent, and 25 -cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$ 7.10$ in postage? (Note: The amount $\$ 7.10$ corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
(A) 45
(B) 46
(C) 51
(D) 54
(E) 55

ANSWER :

(E) 55

PROBLEM 15 :

Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? Here's a hint… if you aren't correct, think about using conversions, maybe that's why you're wrong! -RyanZ4552

(A) 4
(B) 4.2
(C) 4.5
(D) 4.8
(E) 5

ANSWER :

(B) 4.2


PROBLEM 16 :

The letters $\mathrm{P}, \mathrm{Q}$, and R are entered into a $20 \times 20$ table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?

(A) 132 Ps, $134 \mathrm{Qs}, 134 \mathrm{Rs}$
(B) $133 \mathrm{Ps}, 133 \mathrm{Qs}, 134 \mathrm{Rs}$
(C) $133 \mathrm{Ps}, 134 \mathrm{Qs}, 133 \mathrm{Rs}$
(D) $134 \mathrm{Ps}, 132 \mathrm{Qs}, 134 \mathrm{Rs}$
(E) $134 \mathrm{Ps}, 133 \mathrm{Qs}, 133 \mathrm{Rs}$

ANSWER :

(C) $133 \mathrm{Ps}, 134 \mathrm{Qs}, 133 \mathrm{Rs}$

PROBLEM 17 :

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$ ?

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

ANSWER :

(A) 1

PROBLEM 18 :

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5 pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located 2023 pads to the right of her starting position?
(A) 405
(B) 407
(C) 409
(D) 411
(E) 413

ANSWER :

(D) 411

PROBLEM 19 :

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

(A) $1: 3$
(B) $3: 8$
(C) $5: 12$
(D) $7: 16$
(E) $4: 9$

ANSWER :

(C) $5: 12$

PROBLEM 20 :

Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
(A) 56
(B) 57
(C) 58
(D) 60
(E) 61

ANSWER :

(D) 60

PROBLEM 21 :

Alina writes the numbers $1,2, \ldots, 9$ on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

ANSWER :

(C) 2

PROBLEM 22 :

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is 4000 . What is the first term?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 10

ANSWER :

(D) 5

PROBLEM 23 :

Each square in a $3 \times 3$ grid is randomly filled with one of the 4 gray and white tiles shown below on the right.

What is the probability that the tiling will contain a large gray diamond in one of the smaller $2 \times 2$ grids? Below is an example of such tiling.

(A) $\frac{1}{1024}$
(B) $\frac{1}{256}$
(C) $\frac{1}{64}$
(D) $\frac{1}{16}$
(E) $\frac{1}{4}$

ANSWER :

(C) $\frac{1}{64}$

PROBLEM 24 :

Isosceles $\triangle A B C$ has equal side lengths $A B$ and $B C$. In the figure below, segments are drawn parallel to $\overline{A C}$ so that the shaded portions of $\triangle A B C$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle A B C$ ? (Diagram not drawn to scale.)

(A) 14.6
(B) 14.8
(C) 15
(D) 15.2
(E) 15.4

ANSWER :

(A) 14.6

PROBLEM 25 :

Fifteen integers $a_1, a_2, a_3, \ldots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that

$$
1 \leq a_1 \leq 10,13 \leq a_2 \leq 20, \text { and } 241 \leq a_{15} \leq 250 .
$$

What is the sum of digits of $a_{14}$ ?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

SOLUTION :

(A) 8


American Math Competition 8 (AMC 8) 2025 - Problem and Solution

Here are the problems of American Math Competition 8 of the year 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

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 $\cap \cap \cap|\mid$. What number was represented by the following combination of hieroglyphs?

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

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

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

Problem 5

Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labeled $F$ ) 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) 22
(C) 24
(D) 26
(E) 28

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

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

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) \(9 \sqrt{3}\)

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

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

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 $T$

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) $8 \pi$

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?

(A)

(B)

(C)

(D)

(E)

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

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 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

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

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

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) $4 \sqrt{2}$

Problem 19

Two towns, $A$ and $B$, are connected by a straight road, 15 miles long. Traveling from town $A$ 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

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) $\frac{7}{8}$

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

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

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

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

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

What is AMC 8 | How to prepare for AMC 8, 2023-24?

What is AMC 8?

American Mathematics Competition 8 (replaced AJHSME) or AMC 8 is the first step toward International Math Olympiad in the United States (USAMO). Outstanding students participate in this festival of mathematics every year to test their mettle.

This contest aims to provide an opportunity for the students so that they can have a positive attitude towards the challenging problems in Mathematics.

Who can appear for AMC8 2023-24 ?

Students passionate about Mathematics who are in grade 8 or below and under 14.5 years of age on the day of the competition are eligible to participate in the AMC 8.

How can one appear for AMC 8 2023-24 from India?

Cheenta can help you register for American Mathematics Competition (AMC) 8, 2023-24. Interested Students can email us at support@cheenta.com.

Watch this to know why Indian Students should register for AMC.

AMC 8 Exam Format 2023-24

AMC 8 Exam is a 40 minutes Multiple Choice Question Examination. It consists of 25 questions and is available in French, Spanish, Large Print, and Braille, apart from English.

AMC 8 Exam Syllabus 2023-24

The following topics are included in the exam:

The difficulty of the problem increases with the question numbers in the question paper.

Important dates for AMC8, 2023-24

This Exam usually takes place in the month of November on the third or fourth Tuesday of the month. The dates for 2022-2023 are not announced yet. However, you can get an idea of the tentative dates, using the previous year's schedule.

Scoring system of the Examination

After the exam takes place in November, the MAA AMC office will begin emailing official scores and reports in early to mid-December. It takes roughly 3-4 weeks to finish reporting.

What after AMC 8?

The students with high scores in AMC 8, get the chance to participate in AMC 10 by their schools. However, they should be in grade 10 or below.

How to prepare for AMC 8, 2023-24

Resources to prepare for AMC 8 & Curriculum at Cheenta

Preparation Tips

All the best!