MathDB

A5

Part of 2018 Balkan MO Shortlist

Problems(1)

prove that f is a second degree polynomial.

Source: Shortlist BMO 2018, A5

5/3/2019
Let f:R→Rf: \mathbb {R} \to \mathbb {R} be a concave function and g:R→Rg: \mathbb {R} \to \mathbb {R} be a continuous function . If f(x+y)+f(x−y)−2f(x)=g(x)y2 f (x + y) + f (x-y) -2f (x) = g (x) y^2 for all x,y∈R,x, y \in \mathbb {R}, prove that ff is a second degree polynomial.
functional equationalgebrapolynomial