Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied: $NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$
Source: Federal Mathematical Competition of Serbia and Montenegro 2004
Tags: inequalities
Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied: $NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$