MathDB

f(x) - f(y) = a(x, y)(g(x) - g(y))

Source: IMO Shortlist 1992, Problem 12

August 13, 2008

algebranumber theorypolynomialfunctional equationIMO ShortlistIMO Longlist

Problem Statement

Let

f, g

and

a

be polynomials with real coefficients,

f

and

g

in one variable and

a

in two variables. Suppose f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R} Prove that there exists a polynomial

h

with f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.