The Functional Equation that Trolled the NIMO Participants

Algebra Level 5

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