China Western Mathematical Olympiad

No. Suppose the contrary that there are such integers $a$, $b$ and $c$. If one of them is even, say $a$, then $a^{2}bc+2\equiv 2(\mathrm{mod}4)$, which contradicts the assumption that $a^{2}bc+2$ is a perfect square. We may then assume that $a$, $b$ and $c$ are odd, so they are either $1$ or $3(\mathrm{mod}4)$. It follows from the Pigeon-hole Principle that two of them are congruent modulo $4$. Relabel if necessary, we may assume that $a\equiv b(\mathrm{mod}4)$, so $ab{c}^2+2\equiv c^2+2\equiv 1+2\equiv 3(\mathrm{mod}4)$, which violates the perfect square assumption. $\square$