Difference between revisions of "2016 USAMO Problems"

Revision as of 23:57, 26 April 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 $S$ .

Solution

Problem 2

This problem has not been edited in. If you know this problem, please help us out by adding it. 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

This problem has not been edited in. If you know this problem, please help us out by adding it. 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 $I_B F$ and $I_C E$ 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 $y$ , $(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$

Solution

Problem 5

This problem has not been edited in. If you know this problem, please help us out by adding it.

Problem 6

This problem has not been edited in. If you know this problem, please help us out by adding it.

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

2016 USAMO (Problems • Resources)
Preceded by 2015 USAMO	Followed by 2017 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

@@ Line 12: / Line 12: @@
 ===Problem 3===
 {{problem}}
+Let <math>\triangle ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and circumcenter, respectively. Points <math>E</math> and <math>Y</math> are selected on <math>\overline{AC}</math> such that <math>\angle ABY = \angle CBY</math> and <math>\overline{BE}\perp\overline{AC}.</math> Similarly, points <math>F</math> and <math>Z</math> are selected on <math>\overline{AB}</math> such that <math>\angle ACZ = \angle BCZ</math> and <math>\overline{CF}\perp\overline{AB}.</math>
+Lines <math>I_B F</math> and <math>I_C E</math> meet at <math>P.</math> Prove that <math>\overline{PO}</math> and <math>\overline{YZ}</math> are perpendicular.
 ==Day 2==
 ===Problem 4===

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