- Show that for every set of three integers we can find two of them whose average is an integer.
- Show that for every set of five integers we can find three of them whose average is an integer.
(Conjecture) For all k, every set of 2k-1 integers, there exists k of them whose average is an integer.
- The UMCP competition asked for the k=2 and k=3 cases of the conjecture. They are true and easy.
- I have done the k=4 case. It was tedious but not hard.
- I think I have done the k=5 case but it was alot of cases so I may have missed one.
What I hope happens: Someone posts a nice proof.
What would also be okay: Someone posts a counterexample.