f(f(x)) + af(x) = b(a + b)x
Source: IMO Shortlist 1992, Problem 2
August 13, 2008
quadraticsalgebrafunctional equationrecurrence relationIMO Shortlist
Problem Statement
Let \mathbb{R}^\plus{} be the set of all non-negative real numbers. Given two positive real numbers and suppose that a mapping f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{} satisfies the functional equation:
f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.
Prove that there exists a unique solution of this equation.