TSTST 2012 Problem 8

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: Call such a triangular array stable if for every $0 \leq i<j<k \leq n$ we have $$a_{i, j}+a_{j, k} \leq a_{i, k} \leq a_{i, j}+a_{j, k}+1$$ For $s_{1}, \ldots, s_{n}$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_{k}$.