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Geometric Inequalities

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

May 14, 2018
inequalities

Problem Statement

Let M,N,PM, N, P be arbitrary points on the sides BC,CA,ABBC, CA, AB respectively of an acute-angled triangle ABCABC. Prove that at least one of the following inequalities is satisfied: NP≥12BC;PM≥12CA;MN≥12ABNP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB
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