2023 AIME I Problems
2023 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is where and are relatively prime positive integers. Find
Problem 2
Positive real numbers and satisfy the equations The value of is where and are relatively prime positive integers. Find
Problem 3
A plane contains lines, no of which are parallel. Suppose that there are points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, and no points where more than lines intersect. Find the number of points where exactly lines intersect.
Problem 4
The sum of all positive integers such that is a perfect square can be written as where and are positive integers. Find
Problem 5
Let be a point on the circle circumscribing square that satisfies and Find the area of
Problem 6
Alice knows that red cards and black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is where and are relatively prime positive integers. Find
Problem 7
Call a positive integer extra-distinct if the remainders when is divided by and are distinct. Find the number of extra-distinct positive integers less than .
Problem 8
Rhombus has There is a point on the incircle of the rhombus such that the distances from to the lines and are and respectively. Find the perimeter of
Problem 9
Find the number of cubic polynomials where and are integers in such that there is a unique integer with
Problem 10
There exists a unique positive integer for which the sum is an integer strictly between and . For that unique , find .
(Note that denotes the greatest integer that is less than or equal to .)
Problem 11
Find the number of subsets of that contain exactly one pair of consecutive integers. Examples of such subsets are and
Problem 12
Let be an equilateral triangle with side length Points and lie on and respectively, with and Point inside has the property that Find
Problem 13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths and . The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is , where and are relatively prime positive integers. Find . A parallelepiped is a solid with six parallelogram faces such as the one shown below.
Problem 14
The following analog clock has two hands that can move independently of each other. Initially, both hands point to the number . The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock face while the other hand does not move.
Let be the number of sequences of hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the movements, the hands have returned to their initial position. Find the remainder when is divided by .
Problem 15
Find the largest prime number for which there exists a complex number satisfying
- the real and imaginary part of are both integers;
- and
- there exists a triangle whose three side lengths are the real part of and the imaginary part of
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2022 AIME II |
Followed by 2023 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- 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.