I'm tutoring someone in discrete math and one of the questions that came up is this:
Suppose an equilateral triangle made of paper is torn into tiny bits. If the original triangle was 2 meters on a side, show that if you choose 5 of the tiny bits of paper there are 2 bits which were originally no more than 1 meter apart in the triangle.
I don't think this is true as stated. Don't we need to know how many bits the triangle was torn into? If it was torn into, say, 1000 bits, then could we not take 5 bits such that any 2 of them were more than 1 meter apart in the triangle?
I suspect that the question meant to say the triangle was torn into 5 pieces, although I haven't thoroughly pursued this assumption.
Divide the triangle into four smaller triangles of side 1m by joining the midpoints of the sides of the original triangle.
You've chosen 5 tiny pieces and on each of them choose a point. Those points are coming from those 4 smaller triangles, so there must be two coming from the same triangle, and are therefore at most 1m apart.
This is valid no matter how many pieces we had altogether.