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modified version of IMO 2010 SL G3 max { X_iX_{i+1}/P_iP_{i+1}) >=1

Source: 2011 Belarus TST 8.2

November 8, 2020
geometrygeometric inequality

Problem Statement

Let A1A2…AnA_1A_2 \ldots A_n be a convex polygon. Point PP inside this polygon is chosen so that its projections P1,…,PnP_1, \ldots , P_n onto lines A1A2,…,AnA1A_1A_2, \ldots , A_nA_1 respectively lie on the sides of the polygon. Prove that for points X1,…,XnX_1, \ldots , X_n on sides A1A2,…,AnA1A_1A_2, \ldots , A_nA_1 respectively, max⁡{X1X2P1P2,…,XnX1PnP1}≥1.\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1. if a) X1,…,XnX_1, \ldots , X_n are the midpoints of the corressponding sides, b) X1,…,XnX_1, \ldots , X_n are the feet of the corressponding altitudes, c) X1,…,XnX_1, \ldots , X_n are arbitrary points on the corressponding lines.
Modified version of [url=https://artofproblemsolving.com/community/c6h418634p2361975]IMO 2010 SL G3 (it was question c)
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