IMO 1985 LL POL65

Origin: POL

Problem

Define the functions f, F : N \toN, by f(n) = 3 − \sqrt n , F(k) = min{n \inN|f k(n) > 0}, where f k = f ◦\cdot \cdot \cdot ◦f is f iterated n times. Prove that F(k + 2) = 3F(k + 1) −F(k) for all k \inN.