2023 AIME II Problems

2023 AIME II (Answer Key) | AoPS Contest Collections • PDF

Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from to , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, rulers and compasses are permitted. In particular, graph paper, protractors, calculators and computers are not permitted.

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Problem 1

The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is Find the greatest number of apples growing on any of the six trees.

Solution

Problem 2

Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than that is a palindrome both when written in base ten and when written in base eight, such as

Solution

Problem 3

Let be an isosceles triangle with There exists a point inside such that and Find the area of

Solution

Problem 4

Let and be real numbers satisfying the system of equations \begin{align*} xy + 4z &= 60 \\ yz + 4x &= 60 \\ zx + 4y &= 60. \end{align*} Let be the set of possible values of Find the sum of the squares of the elements of

Solution

Problem 5

Let be the set of all positive rational numbers such that when the two numbers and are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of can be expressed in the form where and are relatively prime positive integers. Find

Solution

Problem 6

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 7

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 8

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 9

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 10

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 11

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 12

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 13

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 14

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

Problem 15

These problems will not be available until the 2023 AIME II is released in February 2023.

Solution

See also

• American Invitational Mathematics Examination
• AIME Problems and Solutions
• Mathematics competition resources

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