Difference between revisions of "2016 USAMO Problems"

Revision as of 01:00, 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. 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 lines $MN$ and $PQ.$ 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

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 23: / Line 23: @@
 ===Problem 5===
 {{problem}}
+An equilateral pentagon <math>AMNPQ</math> is inscribed in triangle <math>ABC</math> such that <math>M\in\overline{AB},</math> <math>Q\in\overline{AC},</math> and <math>N, P\in\overline{BC}.</math> Let <math>S</math> be the intersection of lines <math>MN</math> and <math>PQ.</math> Denote by <math>\ell</math> the angle bisector of <math>\angle MSQ.</math>
+Prove that <math>\overline{OI}</math> is parallel to <math>\ell,</math> where <math>O</math> is the circumcenter of triangle <math>ABC,</math> and <math>I</math> is the incenter of triangle <math>ABC.</math>
 ===Problem 6===
 {{problem}}

Page

Toolbox

Search