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Two variable function

Source: Japan Mathematical Olympiad Finals 2018 Q5

February 13, 2018
algebrafunction

Problem Statement

Let TT be a positive integer. Find all functions f:Z+×Z+→Z+f: \mathbb {Z}^+ \times \mathbb {Z}^+ \to \mathbb {Z}^+, such that there exists integers C0,C1,…,CTC_0,C_1,\ldots ,C_T satisfying: (1) For any positive integer nn, the number of positive integer pairs (k,l)(k,l) such that f(k,l)=nf(k,l)=n is exactly nn. (2) For any t=0,1,…,T,t=0,1,\ldots ,T, as well as for any positive integer pair (k,l)(k,l), the equality f(k+t,l+T−t)−f(k,l)=Ctf(k+t,l+T-t)-f(k,l)=C_t holds.
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