Out of the set {1,2,3...100} take any subset of ten numbers and call it T. Prove you can find two disjoint subsets of T such as the sum of the numbers in each subset is the same.

Note: the subsets need not include every number in T; in fact, if you asked for this condition, the problem might be impossible (prove it!).

The second part is simple: if T included an odd number of odd numbers, the total sum of all numbers in T would be odd, and all divisions of T would produce a set with odd sum and other with even sum.

Posted by Oskar on 2004-06-29 13:59:41