¶ 2026 AIME I Problem 5
¶ Problem:
A plane contains points and with . Point is rotated in the plane counterclockwise through an acute angle around point to point . Then is rotated in the plane clockwise through angle around point to point . Suppose . The value of can be written as , where and are relatively prime positive integers. Find .
Announcement
Random Math Together Program - Free for USA(J)MO Qualifiers
Our together program brings together high-achieving students nationwide for team-based problem solving and to prepare for competitions such as HMMT, PUMaC, and more.
Eligibility: USA(J)MO qualifiers • Spots limited
¶ Solution:
Note that forms an angle of with the line , so when we do our second clockwise rotation, we undo the effects of our counterclockwise rotation. Therefore, is simply parallel to , and since it is the same length, is a parallelogram.
Let and , so . Since we have a parallelogram, .
Then
where we use the fact . Thus, we get
for an answer of .
The problems on this page are the property of the MAA's American Mathematics Competitions