Prove that if a prime $p \nmid ijk$ where $i,j,k$ are integers then there exist integers $x, y$, not both divisible by $p$, for which $ix^2 + jy^2 ≡ k (mod\ p)$.
I counted the number of distinct residues $mod\ p$ in the set $\{ix^2(mod\ p), 0\le x\le p-1 \}$ and I found $\frac{p+1}{2}$.
I think that that $ix^2 + jy^2$ ($x,y$ integers) will cover all distinct residues $mod\ p$ but I don't see how to prove it.
Thank you for your help.
The number of distinct residues of the form $k-jy^2$ is also $\frac{p+1}{2}$. This and your set must therefore have at least one element in common!