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p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21

Source: IMO Shortlist 2005 problem A3

July 8, 2006
inequalitiesalgebraIMO Shortlistcalculus

Problem Statement

Four real numbers p p, q q, r r, s s satisfy p+q+r+s=9 p+q+r+s = 9 and p2+q2+r2+s2=21 p^{2}+q^{2}+r^{2}+s^{2}= 21. Prove that there exists a permutation (a,b,c,d) \left(a,b,c,d\right) of (p,q,r,s) \left(p,q,r,s\right) such that ab−cd≥2 ab-cd \geq 2.
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