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An iscoceles triangle

Source: Iran TST 2007, Day 4

May 28, 2007
analytic geometrygeometrytrigonometryinequalitiescircumcirclegeometry proposed

Problem Statement

Triangle ABCABC is isosceles (AB=ACAB=AC). From AA, we draw a line ℓ\ell parallel to BCBC. P,QP,Q are on perpendicular bisectors of AB,ACAB,AC such that PQ⊥BCPQ\perp BC. M,NM,N are points on ℓ\ell such that angles ∠APM\angle APM and ∠AQN\angle AQN are π2\frac\pi2. Prove that 1AM+1AN≤2AB\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}
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