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f (2019)=? g (x^2) = f (x) , f (x + 1) - f (x - 1) = x , g (1) = 0

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2019 3.4

September 21, 2021
algebrafunctionfunctional equation

Problem Statement

Let g:R→Rg: R \to R be a linear function such that g(1)=0g (1) = 0. If f:R→Rf: R \to R is a quadratic function such what g(x2)=f(x)g (x^2) = f (x) and f(x+1)−f(x−1)=xf (x + 1) - f (x - 1) = x for all x∈Rx \in R. Determine the value of f(2019)f (2019).
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