Art of Problem Solving (2024)

Problem 1

Kate bakes a -inch by -inch pan of cornbread. The cornbread is cut into pieces that measure inches by inches. How many pieces of cornbread does the pan contain?

Solution

Problem 2

Sam drove miles in minutes. His average speed during the first minutes was mph (miles per hour), and his average speed during the second minutes was mph. What was his average speed, in mph, during the last minutes?

Solution

Problem 3

In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?

Solution

Problem 4

A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are and square units. What is ?

Solution

Problem 5

How many subsets of contain at least one prime number?

Solution

Problem 6

A box contains chips, numbered and . Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds . What is the probability that draws are required?

Solution

Problem 7

In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is . What is ?

Solution

Problem 8

Sara makes a staircase out of toothpicks as shown:

This is a -step staircase and uses toothpicks. How many steps would be in a staircase that used toothpicks?

Solution

Problem 9

The faces of each of standard dice are labeled with the integers from to . Let be the probability that when all dice are rolled, the sum of the numbers on the top faces is . What other sum occurs with the same probability ?

Solution

Problem 10

In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?

Solution

Problem 11

Which of the following expressions is never a prime number when is a prime number?

Solution

Problem 12

Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

Solution

Problem 13

How many of the first numbers in the sequence are divisible by ?

Solution

Problem 14

A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?

Solution

Problem 15

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?

Solution

Problem 16

Let be a strictly increasing sequence of positive integers such that What is the remainder when is divided by ?

Solution

Problem 17

In rectangle , and . Points and lie on , points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?

Solution

Problem 18

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?

Solution

Problem 19

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is year older than Chloe, and Zoe is exactly year old today. Today is the first of the birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Solution

Problem 20

A function is defined recursively by and for all integers . What is ?

Solution

Problem 21

Mary chose an even -digit number . She wrote down all the divisors of in increasing order from left to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?

Solution

Problem 22

Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?

Solution

Problem 23

How many ordered pairs of positive integers satisfy the equation where denotes the greatest common divisor of and , and denotes their least common multiple?

Solution

Problem 24

Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?

Solution

Problem 25

Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?

Solution

See also

• AMC 10
• AMC 10 Problems and Solutions
• Mathematics competitions
• Mathematics competition resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.