Question 109
Let $N$ be a positive integer. Suppose some collection of integers are written on a blackboard satisfying the following properties.
* Every number $k$ written satisfies $1 \le k \le N$.
* Every $k$ with $1 \le k \le N$ is written at least once.
* The sum of all the numbers written is even.
Prove that by marking some of the numbers written by $\mathcal{O}$ and the rest by $\times$, it is possible to make the sum of those marked by $\mathcal{O}$ equal to the sum of those marked by $\times$.
* Every number $k$ written satisfies $1 \le k \le N$.
* Every $k$ with $1 \le k \le N$ is written at least once.
* The sum of all the numbers written is even.
Prove that by marking some of the numbers written by $\mathcal{O}$ and the rest by $\times$, it is possible to make the sum of those marked by $\mathcal{O}$ equal to the sum of those marked by $\times$.