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A function equation

Source: 2017 Taiwan TST 2nd round day 2 P4

April 15, 2017
functionalgebracombinatoricsnumber theoryfunctional equation

Problem Statement

Find all integer c∈{0,1,...,2016}c\in\{0,1,...,2016\} such that the number of f:Z→{0,1,...,2016}f:\mathbb{Z}\rightarrow\{0,1,...,2016\} which satisfy the following condition is minimal:\\ (1) ff has periodic 20172017\\ (2) f(f(x)+f(y)+1)−f(f(x)+f(y))≡c(mod2017)f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}\\
Proposed by William Chao
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