Let N be postive integer. Some integers are written in a black board and those satisfy the following conditions.
1. Any numbers written are integers which are from 1 to N.
2. More than one integer which is from 1 to N is written.
3. The sum of numbers written is even.
If we mark X to some numbers written and mark Y to all remaining numbers, then prove that we can set the sum of numbers marked X are equal to that of numbers marked Y. inductioncombinatorics proposedcombinatorics