USAMO 2011

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$ . Prove that $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.$$

An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.

In hexagon $ABCDEF$ , which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$ , $\angle C=3\angle F$ , and $\angle E=3\angle B$ . Furthermore $AB=DE$ , $BC=EF$ , and $CD=FA$ . Prove that diagonals $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ are concurrent.

Consider the assertion that for each positive integer $n\geq2$ , the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$ . Either prove the assertion or find (with proof) a counterexample.

Let $P$ be a given point inside quadrilateral $ABCD$ . Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $$\angle Q_1BC=\angle ABP, \angle Q_1CB=\angle DCP, \angle Q_2AD=\angle BAP, \angle Q_2DA=\angle CDP.$$ Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if $\overline{Q_1Q_2}\parallel\overline{CD}$ .

Let $A$ be a set with $|A|=225$ , meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$ . Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$ , and give an example for which equality holds.