I found this problem on Cut The Knot, who got it from [1].
You are given seven distinct positive integers that sum to 100. Prove that some three of them must add up to at 50 or more.
I'll put my solution in the comments later this evening.
[1] Andreescu, T. and Răzvan, G. Mathematical Olympiad Challeges(Burkhäuser, 2004, p 60).