IMO Shortlist 2010 G3

[Stefan Spataru]

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
$\[\max \left\{ \frac{X_1X_2}{P_1P_1}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]$