Mongolian Mathematical Olympiad

If 3 of points lie simultaneously on a line there is nothing to prove. Therefore suppose that, no two of them don't lie on a straight line and . We claim there is a convex polygon with $k\le n$ vertices such that all $n$ points lie inside of the polygon. It is obvious that there exists an angle of the polygon not greater than $\frac{(n-2)\pi }{n}$. Since all points lie inside the angle we can draw rays through all points from vertex of the angle. By the way the angle is divided in $(n-2)$ angles, the sum of which is not less than $\frac{(n-2)\pi }{n}$. It implies that there exists an angle not greater than $\frac{\pi }{n}$. If we take 3 points which form the angle we have done.