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.