BALTIC-WAY 2014 Problem 2

Let $a_{0}, a_{1}, \ldots, a_{N}$ be real numbers satisfying $a_{0}=a_{N}=0$ and

$$a_{i+1}-2 a_{i}+a_{i-1}=a_{i}^{2}$$

for $i=1,2, \ldots, N-1$. Prove that $a_{i} \leqslant 0$ for $i=1,2, \ldots, N-1$.