2011 IMO Shortlist A2

Determine all sequences $\left(x_{1}, x_{2}, \ldots, x_{2011}\right)$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_{1}^{n}+2 x_{2}^{n}+\cdots+2011 x_{2011}^{n}=a^{n+1}+1 .$$

确定正整数的所有序列$\left(x_{1}, x_{2}, \ldots, x_{2011}\right)$，使得对于每个正整数$n$，都有一个整数$a$，其中$$x_{1}^{n}+2 x_{2}^{n}+\cdots+2011 x_{2011}^{n}=a^{n+1}+1 。$$