Nemotron-IMO-Bench 2026 C21

Let $n$ be a positive integer. On each edge of $K_{2^n+1}$ we put an integer in $\{0,1,...,2^{n-1}-1\}$ such that for any triangle the number on some edge of the triangle is the sum of the numbers on the other two edges. Prove that there is a triangle having only zeros on its edges.