2016 USAMO Problems

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 $S$ .

Solution

Problem 2

This problem has not been edited in. Help us out by adding it. Prove that for any positive integer $k,$

\[\left(k^2\right)!\cdot\product_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}\] (Error compiling LaTeX. Unknown error_msg)

is an integer.

Problem 3

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Day 2

Problem 4

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

Solution

Problem 5

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Problem 6

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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

2016 USAMO (Problems • Resources)
Preceded by 2015 USAMO	Followed by 2017 USAMO
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2016 USAMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6