Some Integers written in a black board
Source: 2009 Japan Mathematical Olympiad Finals, Problem 2
February 21, 2009
inductioncombinatorics proposedcombinatorics
Problem Statement
Let 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 .
2. More than one integer which is from 1 to is written.
3. The sum of numbers written is even.
If we mark to some numbers written and mark to all remaining numbers, then prove that we can set the sum of numbers marked are equal to that of numbers marked .