2024 AIME I Problems
2024 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Contents
[hide]Problem 1
Every morning Aya goes for a -kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of
kilometers per hour, the walk takes her
hours, including
minutes spent in the coffee shop. When she walks at
kilometers per hour, the walk takes her
hours and
minutes, including
minutes spent in the coffee shop. Suppose Aya walks at
kilometers per hour. Find the number of minutes the walk takes her, including the
minutes spent in the coffee shop.
Problem 2
There exist real numbers and
, both greater than 1, such that
Find
.
Problem 3
Alice and Bob play the following game. A stack of tokens lies before them. The players take turns with Alice going first. On each turn, the player removes
token or
tokens from the stack. The player who removes the last token wins. Find the number of positive integers
less than or equal to
such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
Problem 4
Jen enters a lottery by selecting distinct elements of
Then four elements of
are drawn at random. Jen wins a prize if at least two of her numbers were drawn, and wins the grand prize if all four of her numbers are drawn. The probability that Jen wins the grand prize given that Jen wins a prize is
where
and
are relatively prime positive integers. Find
.
Problem 5
Rectangle has dimensions
and
, and rectangle
has dimensions
and
. Points
,
,
, and
lie on line
in that order, and
and
lie on opposite sides of line
, as shown. Points
,
,
, and
lie on a common circle. Find
.
Problem 6
Consider the paths of length that follow the lines from the lower left corner to the upper right corner on an
grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.
Problem 7
Find the largest possible real part of where
is a complex number with
. Here
.
Problem 8
Eight circles of radius can be placed tangent to
of
so that the circles are sequentially tangent to each other, with the first circle being tangent to
and the last circle being tangent to
, as shown. Similarly,
circles of radius
can be placed tangent to
in the same manner. The inradius of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 9
Let ,
,
,
be points on the hyperbola
such that
is a rhombus whose diagonals intersect at the origin. Find the largest number less than
for all rhombuses
.
Problem 10
Let have side lengths
,
,
. The tangents to circumcircle of
at
and
intersect at point
, and
intersects the circumcircle at
. The length of
is equal to
, where
and
are relatively prime integers. Find
.
Problem 11
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there had been red vertices is , where
and
are relatively prime positive integers. Find
.
Problem 12
Define and
. Find the number of intersections of the graphs of
and
.
Problem 13
Let be the least prime number for which there exists an integer
such that
is divisible by
. Find the least positive integer
such that
is divisible by
.
Problem 14
Let be a tetrahedron such that
,
, and
. There exists a point
inside the tetrahedron such that the distances from
to each of the faces of the tetrahedron are all equal. This distance can be written in the form
, when
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 15
Let be the set of rectangular boxes with surface area
and volume
. Let
be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of
. The value of
can be written as
, where
and
are relatively prime positive integers. Find
.
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2023 AIME II |
Followed by 2024 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.