Problem: Let $A$ be a set of $6$ points in a plane such that no $3$ are collinear. Show that there exist 3 points in $A$ which form a triangle having an interior angle not $30$ degrees.
I am supposed to use the Pigeonhole Principle for this problem but I am unable to see how to do that here. I considered the triangle with vertices in the point set and least area and observed that no other point in $A$ can be inside this triangle. But I couldn't make use of it.
Also, since the points don't necessarily form a convex set I am unable to use the angle restrictions that we have on convex polygon.
I have a hunch that we need to show that some mean is less than 30 (maybe we can show that the sum of smallest angles taken over all the 20 triangle s is less than 600?)
Someone Please help!