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 that
The 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
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
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.