2012 USAMO Problems
Contents
Day 1
Problem 1
Find all integers such that among any positive real numbers , , , with there exist three that are the side lengths of an acute triangle.
Problem 2
A circle is divided into congruent arcs by points. The points are colored in four colors such that points are colored red, points are colored green, points are colored blue and the remaining points are colored yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
Problem 3
Determine which integers have the property that there exists an infinite sequence , , , of nonzero integers such that the equality holds for every positive integer .
Day 2
Problem 4
Find all functions (where is the set of positive integers) such that for all positive integers and such that divides for all distinct positive integers , .
Problem 5
Let be a point in the plane of triangle , and a line passing through . Let , , be the points where the reflections of lines , , with respect to intersect lines , , , respectively. Prove that , , are collinear.
Problem 6
For integer , let , , , be real numbers satisfying For each subset , define (If is the empty set, then .)
Prove that for any positive number , the number of sets satisfying is at most . For what choices of , , , , does equality hold?
See Also
2012 USAMO (Problems • Resources) | ||
Preceded by 2011 USAMO |
Followed by 2013 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.