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USA Math Olympiad

USAMO 2016

Day 1
Problem 1

Let X_1, X_2, \ldots, X_{100} be a sequence of mutually distinct nonempty subsets of a set S. Any two sets X_i and X_{i+1} are disjoint and their union is not the whole set S, that is, X_i\cap X_{i+1}=\emptyset and X_i\cup X_{i+1}\neq S, for all i\in\{1, \ldots, 99\}. Find the smallest possible number of elements in S.

Problem 2

Prove that for any positive integer k, \left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!} is an integer.

Problem 3

Let \triangle ABC be an acute triangle, and let I_B, I_C, and O denote its B-excenter, C-excenter, and circumcenter, respectively. Points E and Y are selected on \overline{AC} such that \angle ABY = \angle CBY and \overline{BE}\perp\overline{AC}. Similarly, points F and Z are selected on \overline{AB} such that \angle ACZ = \angle BCZ and \overline{CF}\perp\overline{AB}.

Lines \overleftrightarrow{I_B F} and \overleftrightarrow{I_C E} meet at P. Prove that \overline{PO} and \overline{YZ} are perpendicular.

Day 2
Problem 4

Find all functions f:\mathbb{R}\rightarrow \mathbb{R} such that for all real numbers x and y, (f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.

Problem 5

An equilateral pentagon AMNPQ is inscribed in triangle ABC such that M\in\overline{AB}, Q\in\overline{AC}, and N, P\in\overline{BC}. Let S be the intersection of \overleftrightarrow{MN} and \overleftrightarrow{PQ}. Denote by \ell the angle bisector of \angle MSQ.

Prove that \overline{OI} is parallel to \ell, where O is the circumcenter of triangle ABC, and I is the incenter of triangle ABC.

Problem 6

Integers n and k are given, with n\ge k\ge 2. You play the following game against an evil wizard.

The wizard has 2n cards; for each i = 1, ..., n, there are two cards labeled i. 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 k 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 k chosen cards and turns them back face-down. Then, it is your turn again.

We say this game is \textit{winnable} if there exist some positive integer m and some strategy that is guaranteed to win in at most m moves, no matter how the wizard responds.

For which values of n and k is the game winnable?

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

By Ashani Dasgupta

Ph.D. in Mathematics from University of Wisconsin Milwaukee (USA)
Founder - Faculty at Cheenta

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