Prove that among any five points selected inside an equilateral triangle with side equal to one inch, there always exists a pair at the distance not greater than one half an inch.

We can argue using the pigeon-hole principle...

Suppose we divide our equilateral triangle into 4 smaller equilateral triangle regions each of side length one half an inch, by connecting the midpoints of each side:

These smaller regions will be our "pigeon-holes". Now, if one selects $5$ points (our "pigeons") inside the larger triangle, and noting that $5$ is one more than the number of smaller triangular regions -- the pigeon-hole principle requires that 2 of these points be in at least one of the smaller triangles. Since the maximum distance between two points in one of these smaller triangles is the length of the side of that smaller triangle, namely one half an inch, we know there exists a pair of points out of the group of five originally selected that will be at distance not greater than one half an inch from one another.