2010 USAMO Problems
Let be a convex pentagon inscribed in a semicircle of diameter . Denote by , , , the feet of the perpendiculars from onto lines , , , , respectively. Prove that the acute angle formed by lines and is half the size of , where is the midpoint of segment .
There are students standing in a circle, one behind the other. The students have heights . If a student with height is standing directly behind a student with height or less, the two students are permitted to switch places. Prove that it is not possible to make more than such switches before reaching a position in which no further switches are possible.
The 2010 positive numbers satisfy the inequality for all distinct indices . Determine, with proof, the largest possible value of the product .
Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments , , , , , to all have integer side lengths.
Let where is an odd prime, and let
Prove that if for relatively prime integers and , then is divisible by .
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer at most one of the pairs and is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
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