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BMO Round 1, 2020

Algebra Difficulty 7.0 National Olympiad, round 2 Find the answer United Kingdom

A function ff is called good if it assigns an integer value f(m,n)f(m, n) to every ordered pair of integers (m,n)(m, n) in such a way that for every pair of integers (m,n)(m, n) we have:
2f(m,n)=f(m−n,n−m)+m+n=f(m+1,n)+f(m,n+1)−1.2f(m, n) = f(m - n, n - m) + m + n = f(m + 1, n) + f(m, n + 1) - 1.
Find all good functions.

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