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
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 ![\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]](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
Let 
be an equilateral triangle with side length 
Points 
and 
lie on 
and 
respectively, with 
and 
Point inside has the property that ![\[\angle AEP = \angle BFP = \angle CDP.\]](http://latex.artofproblemsolving.com/c/e/b/ceb6f0d1eccf97691c257f73f3128ff839c1f16f.png)
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.
![[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]](http://latex.artofproblemsolving.com/b/c/c/bcc8a67600e8a557236d37e2d01ae1737ef052b5.png)
Problem 14
The following analog clock has two hands that can move independently of each other.
![[asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 1; i < 13; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy]](http://latex.artofproblemsolving.com/f/4/7/f470cd7a8d01e74b5e0dd355c29a6e0b2506c385.png)
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. 
