2010 IMO Shortlist G3

Let $A_{1} A_{2} \ldots A_{n}$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_{1}, \ldots, P_{n}$ onto lines $A_{1} A_{2}, \ldots, A_{n} A_{1}$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_{1}, \ldots, X_{n}$ on sides $A_{1} A_{2}, \ldots, A_{n} A_{1}$ respectively, $$\max \left\{\frac{X_{1} X_{2}}{P_{1} P_{2}}, \ldots, \frac{X_{n} X_{1}}{P_{n} P_{1}}\right\} \geq 1$$ (Armenia)

设 $A_{1} A_{2} \ldots A_{n}$ 为凸多边形。选择该多边形内部的点$P$，使其在直线$A_{1} A_{2}、\ldots、A_{n} A_{1}$上的投影$P_{1}、\ldots、P_{n}$分别位于多边形的两侧。证明对于边 $A_{1} A_{2}、\ldots、A_{n} A_{1}$ 上的任意点 $X_{1}, \ldots, X_{n}$ 分别有 $$\max \left\{\frac{X_{1} X_{2}}{P_{1} P_{2}}, \ldots, \frac{X_{n} X_{1}}{P_{n} P_{1}}\right\} \geq 1$$(亚美尼亚)