2025 AIME I Problems
2025 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Contents
[hide]Problem 1
Find the sum of all integer bases for which
is a divisor of
.
Problem 2
On points
and
lie on
so that
, while points
and
lie on
so that
. Suppose
,
,
,
,
, and
. Let
be the reflection of
through
, and let
be the reflection of
through
. The area of quadrilateral
is
. Find the area of heptagon
, as shown in the figure below.
Problem 3
The members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when
is divided by
Problem 4
Find the number of ordered pairs , where both
and
are integers between
and
inclusive, such that
.
Problem 5
There are eight-digit positive integers that use each of the digits
exactly once. Let
be the number of these integers that are divisible by
. Find the difference between
and
.
Problem 6
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths and
, with
. Find
.
Problem 7
The twelve letters ,
,
,
,
,
,
,
,
,
,
, and
are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is
,
,
,
,
,
. The probability that the last word listed contains
is
, where
and
are relatively prime positive integers. Find
.
Problem 8
Let be a real number such that the system
. The sum of all possible values of
can be written as
, where
and
are relatively prime positive integers. Find
. Here
.
Problem 9
The parabola with equation is rotated
counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has
-coordinate
, where
,
, and
are positive integers, and
and
are relatively prime. Find
.
Problem 10
The cells of a
grid are filled in using the numbers
through
so that each row contains
different numbers, and each of the three
blocks heavily outlined in the example below contains
different numbers, as in the first three rows of a Sudoku puzzle.
The number of different ways to fill such a grid can be written as where
,
,
, and
are distinct prime numbers and
,
,
,
are positive integers. Find
.
Problem 11
A piecewise linear function is defined by and
for all real numbers
. The graph of
has a sawtooth pattern.
The parabola intersects the graph of
at finitely many points. The sum of the
-coordinates of all these intersection points can be expressed in the form
, where
,
,
, and
are positive integers such that
,
,
have greatest common divisor equal to
, and
is not divisible by the square of any prime. Find
.
Problem 12
The set of points in -dimensional coordinate space that lie in the plane
whose coordinates satisfy the inequalities
forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 13
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these
line segments divide the disk.
Problem 14
Let be a convex pentagon with
and
For each point
in the plane, define
The least possible value of
can be expressed as
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 15
Let denote the number of ordered triples of positive integers
such that
and
is a multiple of
. Find the remainder when
is divided by
.
See also
2025 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2024 AIME II |
Followed by 2025 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.