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Inequality on a function

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #1

January 21, 2017
inequalitiesfunction

Problem Statement

Let f:Z→Rf : \mathbb{Z} \to \mathbb{R} be a function satisfying f(x)+f(y)+f(z)≥0f(x) + f(y) + f(z) \ge 0 for all integers x,y,zx, y, z with x+y+z=0x + y + z = 0. Prove that f(−2017)+f(−2016)+⋯+f(2016)+f(2017)≥0. f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0.
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