Here's a question I think is hard.
Consider a square S of side a in the plane. How many, at most, points P,Q of the square can be chosen either within S or on the sides of S, such that their distance $|P-Q|\geq \sqrt{a}$?
This question troubles me because I can't grasp what I am to prove. If this was a Computer Science question, I would do a simulation for some values of a and a large number of random points to get the answer as a probability. But this is a Combinatorics question, and I do not get what does "how many" imply.
I mean, if it means cardinality, then we have this set obviously has the power of the continuum (as the plane has this property). Then how can it be compared to S?
If the answer involves area, then the condition $|P-Q|\geq a$ constructs a subregion of S, but this will be hard to describe algebraically...
A hint:
This has nothing to do with infinity but with discrete geometry, or more precisely: with circle packings in the plane. The densest circle packing is the hexagonal packing. It gives you an upper bound, but you have to take care of boundary effects.