Let $p\equiv 3\pmod 4$ and $G$ be the set of nonzero quadratic residues modulo $p$ (so $G=(p-1)/2$). For $1\leq a\leq p-1$, let $G_a=\{(a+g)\pmod p\mid g\in G\}$. What is the size of $G_0\cap G_a$?
I tested for $p=3,7$ and $11$, and the size of the intersection is always $(p-3)/4$. Is this always true? If so, it sounds like it should be a basic fact of quadratic residues, but I couldn't find it.