2012 IMO Shortlist N6

Let $x$ and $y$ be positive integers. If $x^{2^{n}}-1$ is divisible by $2^{n} y+1$ for every positive integer $n$, prove that $x=1$.

令$x$ 和$y$ 为正整数。如果对于每个正整数 $n$，$x^{2^{n}}-1$ 都能被 $2^{n} y+1$ 整除，则证明 $x=1$。