USAJMO 2011

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

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

For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$ , such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$ . Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.

A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$ , and for $n \ge 2$ , $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$ . Prove that for any $n \ge 1$ , the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.

Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$ , (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$ . Prove that $BE$ bisects $AC$ .

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.