NO-ABEL 1997

We call a positive integer $n$ happy if there exist integers $a,b$ such that $a^2+b^2 = n$ . If $t$ is happy, show that (a) $2t$ is happy, (b) $3t$ is not happy

Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$ . Show that the equation $f(x) = x$ has at most one integer solution.