Question 70
Determine all sequences of positive integers $\{x_1,x_2,\ldots,x_{101}\}$ such that for every positive integer $n$ there is an integer $a$ satisfying
$$x_1^n+2x_2^n+\cdots+101x_{101}^n=a^{n+1}+1.$$
$$x_1^n+2x_2^n+\cdots+101x_{101}^n=a^{n+1}+1.$$