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largest real constant c, such p^2 + q^2 + 1 >= bp(q + 1(

Source: Austrian - Polish 1997 APMC

May 3, 2020
inequalitiesmaxalgebra

Problem Statement

(a) Prove that p2+q2+1>p(q+1)p^2 + q^2 + 1 > p(q + 1) for any real numbers p,qp, q, . (b) Determine the largest real constant bb such that the inequality p2+q2+1≥bp(q+1)p^2 + q^2 + 1 \ge bp(q + 1) holds for all real numbers p,qp, q (c) Determine the largest real constant c such that the inequality p2+q2+1≥cp(q+1)p^2 + q^2 + 1 \ge cp(q + 1) holds for all integers p,qp, q.
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