2023 AIME I Problems
| 2023 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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 ![\[\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).\]](http://latex.artofproblemsolving.com/e/1/8/e188ed1d68418cfaed35d91cc7b0f333a84f6b1b.png)
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
There is a rhombus in which 
. A point is chosen somewhere on the incircle of , and the distances from to sides 
, 
, and 
, are 
, 
, and , respectively. Evaluate the perimeter of the rhombus.
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 ![\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]](http://latex.artofproblemsolving.com/6/b/0/6b010978a55642b7f7aabb2df2f7cf45c9db937a.png)
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
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
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
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 15
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
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. 
