The Functional Equation that Trolled the NIMO Participants

Algebra Level 5

Let f f be a non-constant polynomial such that f(x1)+f(x)+f(x+1)=f(x)22013x f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} for all nonzero real numbers x x . Let N N the sum of all possible values of f(1) f(1) . Find the remainder when N N is divided by 1000 1000 .

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