26th Hellenic Mathematical Olympiad

We have $$K=\frac{9n^2+31}{n^2+7}=\frac{9(n^2+7)-32}{n^2+7}=9-\frac{32}{n^2+7}.$$ Since $K$ is integer, it follows that $n^2+7$ is a divisor of $32$ and taking in mind that $n^2+7\ge 8$, we conclude: $$n^2+7\in \{8,16,32\}\iff n^2\in \{1,9,25\}\iff n\in \{-1,1,-3,3,-5,5\}.$$