Selected Problems from Open Contests

Adding up all equations gives $p^2-cp+q^2-aq+r^2-br=-12$. From the inequality $(p-\frac{c}{2}{)}^2\ge 0$ we have $p^2-cp\ge -\frac{c^2}{4}\ge -4$ and similarly, $q^2-aq\ge -4$ and $r^2-br\ge -4$. Adding up these inequalities, we see that to avoid a contradiction with the equality derived first, all three inequalities must actually be equalities, i.e. $a=b=c=4$ and $p=q=r=2$. But this does not satisfy the initial equations.