2011 IMO Shortlist N6

Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x)$. Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1$. Prove that $Q(x)$ is a constant polynomial.

令 $P(x)$ 和 $Q(x)$ 为两个具有整数系数的多项式，使得具有有理系数的非常数多项式不能同时整除 $P(x)$ 和 $Q(x)$。假设对于每个正整数 $n$，整数 $P(n)$ 和 $Q(n)$ 都是正数，并且 $2^{Q(n)}-1$ 除以 $3^{P(n)}-1$。证明 $Q(x)$ 是常数多项式。