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inequality, hidden after a geo cover

Source: Romanian ROM TST 2004, problem 1, created by Calin Popescu

May 1, 2004
inequalitiesgeometryperimeterLaTeXinequalities solved

Problem Statement

Let a1,a2,a3,a4a_1,a_2,a_3,a_4 be the sides of an arbitrary quadrilateral of perimeter 2s2s. Prove that ∑i=141ai+s≤29∑1≤i<j≤41(s−ai)(s−aj). \sum\limits^4_{i=1} \dfrac 1{a_i+s} \leq \dfrac 29\sum\limits_{1\leq i<j\leq 4} \dfrac 1{ \sqrt { (s-a_i)(s-a_j)}}. When does the equality hold?
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