2016 USAMO Problems
Let be a sequence of mutually distinct nonempty subsets of a set . Any two sets and are disjoint and their union is not the whole set , that is, and , for all . Find the smallest possible number of elements in .
Prove that for any positive integer is an integer.
Let be an acute triangle, and let and denote its -excenter, -excenter, and circumcenter, respectively. Points and are selected on such that and Similarly, points and are selected on such that and
Lines and meet at Prove that and are perpendicular.
Find all functions such that for all real numbers and ,
An equilateral pentagon is inscribed in triangle such that and Let be the intersection of and Denote by the angle bisector of
Prove that is parallel to where is the circumcenter of triangle and is the incenter of triangle
Integers and are given, with You play the following game against an evil wizard.
The wizard has cards; for each there are two cards labeled Initially, the wizard places all cards face down in a row, in unknown order.
You may repeatedly make moves of the following form: you point to any of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the chosen cards and turns them back face-down. Then, it is your turn again.
We say this game is winnable if there exist some positive integer and some strategy that is guaranteed to win in at most moves, no matter how the wizard responds.
For which values of and is the game winnable?
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