Question bank USAJMO 2011

Find all positive integers $n$ such that $2^n+12^n+2011^n$ is a perfect square.

Proposed by Titu Andreescu (credit in the supplied notes).

Let $a$, $b$, $c$ be positive real numbers
such that $a^2+b^2+c^2+(a+b+c)^2 \le 4$. Prove that

\[\frac{ab+1}{(a+b)^2} + \frac{bc+1}{(b+c)^2} + \frac{ca+1}{(c+a)^2} \ge 3.\]

Proposed by Titu Andreescu (credit in the supplied notes).

For $P=(a,a^2)$ let $\ell(P)$ be the line through $P$ with slope $2a$. Consider triples of distinct points $P_i=(a_i,a_i^2)$ whose three lines $\ell(P_i)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$.

Proposed by Zuming Feng (credit in the supplied notes).

A word is a finite string of letters; it is a palindrome if it reads identically forwards and backwards. Define $W_0=a$, $W_1=b$ and $W_n=W_{n-2}W_{n-1}$ for $n\ge2$, where juxtaposition means concatenation. Prove that $W_1W_2\cdots W_n$ is a palindrome for every $n\ge1$.

Proposed by Gabriel Carroll (credit in the supplied notes).

Points $A,B,C,D,E$ lie on a circle $\omega$ and $P$ lies outside it. Lines $PB,PD$ are tangent to $\omega$; $P,A,C$ are collinear; and $DE\parallel AC$. Prove that $BE$ bisects the segment $AC$.

Proposed by Zuming Feng (credit in the supplied notes).

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.

Proposed by Sam Vandervelde (credit in the supplied notes).