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min of L/l, max side inscribed in triangle, min square circumscribed in triangle

Source: 2018 Cono Sur Shortlist G5

August 25, 2021
geometryratioinscribedsquarecircumscribedGeometric Inequalities

Problem Statement

We say that a polygon PP is inscribed in another polygon QQ when all the vertices of PP belong to the perimeter of QQ. We also say in this case that QQ is circumscribed to PP. Given a triangle TT, let ℓ\ell be the largest side of a square inscribed in TT and LL is the shortest side of a square circumscribed to TT . Find the smallest possible value of the ratio L/ℓL/\ell .
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