Question 113
Let $m$ be a positive integer, and let real numbers $a_1, a_2, \dots, a_m$ satisfy
$$
\frac{1}{m} \sum_{i=1}^{m} a_i = 1,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^2 = 11,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^3 = 1,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^4 = 131.
$$
Prove that $m$ must be a multiple of 7.
$$
\frac{1}{m} \sum_{i=1}^{m} a_i = 1,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^2 = 11,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^3 = 1,
$$
$$
\frac{1}{m} \sum_{i=1}^{m} a_i^4 = 131.
$$
Prove that $m$ must be a multiple of 7.