2022 AIME I Problems
2022 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Quadratic polynomials and
have leading coefficients
and
respectively. The graphs of both polynomials pass through the two points
and
Find
Problem 2
Find the three-digit positive integer whose representation in base nine is
where
,
, and
are (not necessarily distinct) digits.
Problem 3
In isosceles trapezoid parallel bases
and
have lengths
and
respectively, and
The angle bisectors of
and
meet at
and the angle bisectors of
and
meet at
Find
Problem 4
Let and
where
Find the number of ordered pairs
of positive integers not exceeding
that satisfy the equation
Problem 5
A straight river that is meters wide flows from west to east at a rate of
meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of
meters downstream from Sherry. Relative to the water, Melanie swims at
meters per minute, and Sherry swims at
meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find
Problem 6
Find the number of ordered pairs of integers such that the sequence
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Problem 7
Let be distinct integers from
to
The minimum possible positive value of
can be written as
where
and
are relatively prime positive integers. Find
Problem 8
Equilateral triangle is inscribed in circle
with radius
Circle
is tangent to sides
and
and is internally tangent to
Circles
and
are defined analogously. Circles
and
meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of
are the vertices of a large equilateral triangle in the interior of
and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of
The side length of the smaller equilateral triangle can be written as
where
and
are positive integers. Find
Problem 9
Ellina has twelve blocks, two each of red (), blue (
), yellow (
), green (
), orange (
), and purple (
). Call an arrangement of blocks
if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is
where
and
are relatively prime positive integers. Find
Problem 10
Three spheres with radii
and
are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at
and
respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that
Find
Problem 11
Let be a parallelogram with
A circle tangent to sides
and
intersects diagonal
at points
and
with
as shown. Suppose that
and
Then the area of
can be expressed in the form
where
and
are positive integers, and
is not divisible by the square of any prime. Find
Problem 12
For any finite set let
denote the number of elements in
Define
where the sum is taken over all ordered pairs
such that
and
are subsets of
with
For example,
because the sum is taken over the pairs of subsets
giving
Let
where
and
are relatively prime positive integers. Find the remainder when
is divided by
Problem 13
Let be the set of all rational numbers that can be expressed as a repeating decimal in the form
where at least one of the digits
or
is nonzero. Let
be the number of distinct numerators obtained when numbers in
are written as fractions in lowest terms. For example, both
and
are counted among the distinct numerators for numbers in
because
and
Find the remainder when
is divided by
Problem 14
Given and a point
on one of its sides, call line
the
of
through
if
passes through
and divides
into two polygons of equal perimeter. Let
be a triangle where
and
and
are positive integers. Let
and
be the midpoints of
and
respectively, and suppose that the splitting lines of
through
and
intersect at
Find the perimeter of
Problem 15
Let
and
be positive real numbers satisfying the system of equations:
Then
can be written as
where
and
are relatively prime positive integers. Find
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2021 AIME II |
Followed by 2022 AIME II | |
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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.