2022 AIME I Problems
| 2022 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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 ![\[3,4,5,a,b,30,40,50\]](http://latex.artofproblemsolving.com/1/0/f/10f3618dca767ca85f66cb8fd4b671468c6e2317.png)
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 ![\[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\]](http://latex.artofproblemsolving.com/c/6/5/c653d108c03120ef617df93994006dd9a956c44c.png)
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 ([b]R[/b]), blue ([b]B[/b]), yellow ([b]Y[/b]), green ([b]G[/b]), orange ([b]O[/b]), and purple ([b]P[/b]). Call an arrangement of blocks [i]even[/i] if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
[b]R B B Y G G Y R O P P O[/b]
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
Solution
Problem 12
For any finite set 
let 
denote the number of elements in 
Define ![\[S_n = \sum |A \cap B|,\]](http://latex.artofproblemsolving.com/3/2/7/327537a8b4ed3adcbc11eda448fb51b1deb3e658.png)
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 ![\[(A,B) \in \left\{(\emptyset,\emptyset),(\{1\},\{1\}),(\{1\},\{2\}),(\{2\},\{1\}),(\{2\},\{2\}),(\{1,2\},\{1,2\})\right\},\]](http://latex.artofproblemsolving.com/9/f/1/9f16349df81aa783a674d60e5ddc799f0cb56410.png)
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 [i]splitting line[/i] 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
\begin{align*}
\sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\
\sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\
\sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3.
\end{align*}
Then ![$\left[ (1-x)(1-y)(1-z) \right]^2$](http://latex.artofproblemsolving.com/e/e/1/ee1ff4ecc5f5319e77a3c7a3d7e9534a9ec343da.png)
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 | |
| 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. 
