Question 50
Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows:
Call such a triangular array *stable* if for every $0 \le i \lt j \lt k \le n$ we have
$$
a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1.
$$
For $s_1, \dots, s_n$ any nondecreasing sequence of nonnegative integers, prove there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.
Call such a triangular array *stable* if for every $0 \le i \lt j \lt k \le n$ we have
$$
a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1.
$$
For $s_1, \dots, s_n$ any nondecreasing sequence of nonnegative integers, prove there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.