USAJMO

The answer is $n=1$ , which is easily verified to be a valid integer $n$ .
Notice that $$2^n+12^n+2011^n\equiv 2^n+7^n \pmod{12}.$$ Then for $n\geq 2$ , we have $2^n+7^n\equiv 3,5 \pmod{12}$ depending on the parity of $n$ . But perfect squares can only be $0,1,4,9\pmod{12}$ , contradiction. Therefore, we are done. $\blacksquare$