CMO 2016 Problem 3

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent:

1. There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$ , $$\sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$