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Find the smallest positive real number

Source: IMO Shortlist 2007, G6 / USA TST 2008, Day 2, Problem 6

July 13, 2008
geometrycircumcirclesymmetryratioIMO Shortlist

Problem Statement

Determine the smallest positive real number k k with the following property. Let ABCD ABCD be a convex quadrilateral, and let points A1 A_1, B1 B_1, C1 C_1, and D1 D_1 lie on sides AB AB, BC BC, CD CD, and DA DA, respectively. Consider the areas of triangles AA1D1 AA_1D_1, BB1A1 BB_1A_1, CC1B1 CC_1B_1 and DD1C1 DD_1C_1; let S S be the sum of the two smallest ones, and let S1 S_1 be the area of quadrilateral A1B1C1D1 A_1B_1C_1D_1. Then we always have kS1≥S kS_1\ge S.
Author: Zuming Feng and Oleg Golberg, USA
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