IMO 1989 LL USA106

Origin: USA

Problem

Let n > 1 be a fixed integer. Define functions f0(x) = 0, f1(x) = 1 −cos x, and for k > 0, fk+1(x) = 2fk(x) cos x −fk−1(x). If F(x) = f1(x) + f2(x) + \cdot \cdot \cdot + fn(x), prove that (a) 0 < F(x) < 1 for 0 < x < \pi n+1, and (b) F(x) > 1 for \pi n+1 < x < \pi n.