Question 37
Let $A_1\cdots A_n$ be a convex polygon and $P$ an interior point whose perpendicular projections $P_i$ on $A_iA_{i+1}$ lie inside the respective segments (cyclic indices). Prove that for arbitrary points $X_i$ inside the respective segments, the following source inequality holds:
$$\max\left\{\frac{X_1X_2}{P_1P_2}+\cdots+\frac{X_nX_1}{P_nP_1}\right\}\ge1.$$
$$\max\left\{\frac{X_1X_2}{P_1P_2}+\cdots+\frac{X_nX_1}{P_nP_1}\right\}\ge1.$$