Let
be a convex polygon. Point
inside this polygon is chosen so that its projections
onto lines
respectively lie on the sides of the polygon. Prove that for arbitrary points
on sides
respectively,
Proposed by Nairi Sedrakyan, Armenia
%V0
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,
$$\max \left\{ \frac{X_1X_2}{P_1P_1}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.$$
Proposed by Nairi Sedrakyan, Armenia
Školjka