Adding up all equations gives $p^2 - c p + q^2 - a q + r^2 - b r = -12$. From the inequality $(p - \frac{c}{2})^2 \ge 0$ we have $p^2 - c p \ge -\frac{c^2}{4} \ge -4$ and similarly, $q^2 - a q \ge -4$ and $r^2 - b r \ge -4$. 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.