Question 142
Let $n$ be an integer greater than $2$. We want to color red exactly $n+1$ of the numbers $1, 2, \cdots, 2n-1, 2n$ in such a way that there are no three distinct red numbers $x, y, z$ satisfying the equality $x+y=z$. Prove that there is one and only one way to choose the numbers to be colored red that satisfies the given condition.