IMO 1988 LL INA44

(a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the

imolonglistmathematicsolympiad

IMO 1988 LL INA44

Origin: INA

Problem

(a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the polynomial g(x12) is divided by the polynomial g(x)? (b) If k is a positive integer and f is a function such that for every positive number x, f(x2 +1) \sqrtx = k, find the value of f  9+y2 y2 \sqrt 12/y for every positive number y. (c) The function f satisfies the functional equation f(x) + f(y) = f(x + y) −xy −1 for every pair x, y of real numbers. If f(1) = 1, find the number of integers n for which f(n) = n.