2022 AIME II Problems
2022 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Adults made up of the crowd of people at a concert. After a bus carrying more people arrived, adults made up of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
Problem 2
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability . Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is , where and are relatively prime positive integers. Find .
Problem 3
A right square pyramid with volume has a base with side length The five vertices of the pyramid all lie on a sphere with radius , where and are relatively prime positive integers. Find .
Problem 4
There is a positive real number not equal to either or such thatThe value can be written as , where and are relatively prime positive integers. Find .
Problem 5
Twenty distinct points are marked on a circle and labeled through in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original points.
Problem 6
Let be real numbers such that and . Among all such -tuples of numbers, the greatest value that can achieve is , where and are relatively prime positive integers. Find .
Problem 7
A circle with radius is externally tangent to a circle with radius . Find the area of the triangular region bounded by the three common tangent lines of these two circles.
Problem 8
Find the number of positive integers whose value can be uniquely determined when the values of , , and are given, where denotes the greatest integer less than or equal to the real number .
Problem 9
Let and be two distinct parallel lines. For positive integers and , distinct points lie on , and distinct points lie on . Additionally, when segments are drawn for all and , no point strictly between and lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when and . The figure shows that there are 8 regions when and .
Problem 10
Find the remainder whenis divided by .
Problem 11
Let be a convex quadrilateral with and such that the bisectors of acute angles and intersect at the midpoint of Find the square of the area of
Problem 12
Let and be real numbers with and such thatFind the least possible value of
Problem 13
There is a polynomial with integer coefficients such thatholds for every Find the coefficient of in .
Problem 14
For positive integers , , and with , consider collections of postage stamps in denominations , , and cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to cents, let be the minimum number of stamps in such a collection. Find the sum of the three least values of such that for some choice of and .
Problem 15
Two externally tangent circles and have centers and , respectively. A third circle passing through and intersects at and and at and , as shown. Suppose that , , , and is a convex hexagon. Find the area of this hexagon.
See also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2022 AIME I |
Followed by 2023 AIME I | |
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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.