BALTIC-WAY 2017 Problem 20

Let $S$ be the set of all ordered pairs $(a, b)$ of integers with $0<2 a<2 b<2017$ such that $a^{2}+b^{2}$ is a multiple of 2017. Prove that

$$\sum_{(a, b) \in S} a=\frac{1}{2} \sum_{(a, b) \in S} b$$