Let
![A_1A_2 \ldots A_n](/media/m/3/d/b/3dbfe0f5b1f84bfc9be4ee4d64979438.png)
be a convex polygon. Point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside this polygon is chosen so that its projections
![P_1, \ldots , P_n](/media/m/e/d/8/ed8fb0940ebcb98818b3186dc553a3f6.png)
onto lines
![A_1A_2, \ldots , A_nA_1](/media/m/1/0/d/10d4620246eca2c4e000bd70bb2411f5.png)
respectively lie on the sides of the polygon. Prove that for arbitrary points
![X_1, \ldots , X_n](/media/m/f/b/a/fba6b5a4107a02ec109521a983ac9e65.png)
on sides
![A_1A_2, \ldots , A_nA_1](/media/m/1/0/d/10d4620246eca2c4e000bd70bb2411f5.png)
respectively,
![\max \left\{ \frac{X_1X_2}{P_1P_1}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.](/media/m/1/f/0/1f0f2a912acbb9c38f086cae0409e077.png)
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