2024 AIME II Problems
2024 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Among the residents of Aimeville, there are
who own a diamond ring,
who own a set of golf clubs, and
who own a garden spade. In addition, each of the
residents owns a bag of candy hearts. There are
residents who own exactly two of these things, and
residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
Problem 2
A list of positive integers has the following properties:
The sum of the items in the list is
.
The unique mode of the list is
.
The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
Problem 3
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is , and the sum of the three numbers formed by reading top to bottom is
. The grid below is an example of such an arrangement because
and
.
Problem 4
Let and
be positive real numbers that satisfy the following system of equations:
Then the value of
is
where
and
are relatively prime positive integers. Find
.
Problem 5
Let be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments
,
, and
has side lengths
and
. Find the side length of the hexagon.
Problem 6
Alice chooses a set of positive integers. Then Bob lists all finite nonempty sets
of positive integers with the property that the maximum element of
belongs to
. Bob's list has
sets. Find the sum of the elements of
.
Problem 7
Let be the greatest four-digit integer with the property that whenever one of its digits is changed to
, the resulting number is divisible by
. Let
and
be the quotient and remainder, respectively, when
is divided by
. Find
.
Problem 8
Problem 9
Problem 10
Let have incenter
, circumcenter
, inradius
, and circumradius
. Suppose that
. Find
.
Problem 11
Find the number of triples of nonnegative integers satisfying
and
Problem 12
Problem 13
Let be a 13th root of unity. Find the remainder when
is divided by 1000.
Problem 14
Let be an integer. Call a positive integer
if it has exactly two digits when expressed in base
, and these two digits sum to
. For example,
is
-eautiful because
and
. Find the least integer
for which there are more than ten
-eautiful integers.
Problem 15
Find the number of rectangles inside a fixed regular dodecagon (-gon) where each side of the rectangle lies on a side or on a diagonal of the dodecagon. The diagram below shows three of those rectangles.
See also
2024 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2024 AIME I |
Followed by 2025 AIME I | |
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.