IT-CESENATICO 2019 Problem 3

Let $n>2$ be an integer $.$ We want to color in red exactly $n+1$ of the numbers $1,2,\cdots,2n-1, 2n$ so that there do not exists three distinct red integers $x,y,z$ satisfying $x+y=z.$ Prove that there is one and one only way to color the red numbers according to the given condition $.$