42nd United States of America Mathematical Olympiad
Day I

12:30 PM – 5 PM EDT
April 30, 2013

Note: For any geometry problem, the first page of the solution must be a large, in-scale, clearly labeled diagram made with drawing instruments (ruler, compass, protractor, graph paper). Failure to meet any of these requirements will result in a 1-point automatic deduction.

USAMO 1. In triangle ABC, points P, Q, R lie on sides BC, CA, AB, respectively. Let ωA , ωB , ωC denote the circumcircles of triangles AQR, BRP, CP Q, respectively. Given the fact that segment AP intersects ωA , ωB , ωC again at X, Y, Z respectively, prove that Y X/XZ =
BP/P C.
USAMO 2. For a positive integer n ≥ 3 plot n equally spaced points around a circle. Label one of them
A, and place a marker at A. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of 2n distinct moves available; two from each point. Let an count the number the number of ways to advance around the circle exactly twice, beginning and ending at A, without repeating a move. Prove that an−1 + an = 2n for all n ≥ 4.
USAMO 3. Let n be a positive integer. There are n(n+1) marks, each with a black side and a white side,
2
arranged into an equilateral triangle, with the biggest row containing n marks. Initially, each mark has the black side up. An operation is to choose a line parallel to one of the sides of the triangle, and flipping all the marks on that line. A