National Competition

Set $u:=\frac{ab}{c}$, $v:=\frac{ac}{b}$ and $w:=\frac{bc}{a}$. By assumption, $u+v+w$ is an integer. It is easily seen that $uv+uw+vw=a^2+b^2+c^2$ and $uvw=abc$ are integers, too.

According to Vieta's formulae, the rational numbers $u,v,w$ are the roots of a cubic polynomial $x^3+p{x}^2+qx+r$ with integer coefficients. As the leading coefficient is 1, these roots are integers.