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cos^2φ+q \sinφ \cosφ +r\sin^2φ \ge 1/2 (p+r-\sqrt{(p-r)^2+q^2})

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1991 p1

February 11, 2020
trigonometryinequalities

Problem Statement

Prove that for any real numbers p,q,r,ϕp,q,r,\phi,: cos⁡2ϕ+qsin⁡ϕcos⁡ϕ+rsin⁡2ϕ≥12(p+r−(p−r)2+q2)\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})
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