AUSTRIAN-POLISH 1997 Problem 7

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$ , . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$ .