IMO Shortlist 2007 problem G6


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2. travnja 2012.
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Determine the smallest positive real number k with the following property. Let ABCD be a convex quadrilateral, and let points A_1, B_1, C_1, and D_1 lie on sides AB, BC, CD, and DA, respectively. Consider the areas of triangles AA_1D_1, BB_1A_1, CC_1B_1 and DD_1C_1; let S be the sum of the two smallest ones, and let S_1 be the area of quadrilateral A_1B_1C_1D_1. Then we always have kS_1\ge S.

Author: unknown author, USA
Izvor: Međunarodna matematička olimpijada, shortlist 2007