USSR Problem Book, #248 hint

The USSR Olympiad Problem Book

#248. Prove that, for any prime \(p\), it is possible to find integers \(x\) and \(y\) such that \(x^2 + y^2 + 1\) is divisible by \(p\).

Hint. Choose a small prime (\(p=11\) for example) and look at the set of all squares mod \(p\). Find a pattern.