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Here's the problem that I got first:

Given five points on a plane such that no three of the points are collinear. Show that among the triangles which are drawn using any three of these five points as vertices, at least three of the triangles formed are not acute-angled triangles.

My attempt:

I tried separating into two cases: The five points are a convex set or not.

If the five points are a convex set, then its convex hull must be some five sided convex polygon, meaning that at least one angle is larger than $108^\circ$. (Pigeonhole principle)

But then what should I do next? I have no idea.


Then I got the second version of the problem:

Given any 100 points on a plane such that no three of the points are collinear. Show that among the triangles which are drawn using any three of these 100 points as vertices, at least 30% of the triangles are not acute-angled triangles.

This seems even harder, but I think if I can solve the above, it would be pretty easy to solve this version as well.

Can anyone please help? This is very hard for me, and I really want to know the solution.

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