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2023 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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.