Difference between revisions of "2016 USAMO Problems"

Revision as of 00:57, 27 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. 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. 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. Help us out by adding it.

Problem 6

This problem has not been edited in. Help us out by adding it.

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
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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