... divides (a, (a, b)c) This finishes the proof.u 243 Theorem (a2 , b2 ) = (a, b)2 Proof: Assume that (m, n) = Using the preceding lemma twice, (m2 , n2 ) = (m2 , (m2 , n)n) = (m2 , (n, (m, n)m)n) As (m, ... Problem Let F0 (x) = x, F(x) = 4x(1 − x), Fn+1 (x) = F(Fn (x)), n = 0, 1, Prove that Fn (x) dx = 40 Problem (IMO 1977) Let f , f : N → N be a function satis- fying f (n + 1) > f ( f (n)) for each ... (1 − 2)2n = (1 + 2)2 (1 + 2)2n−2 + (1 − 2)2 (1 − 2)2n−2 This simplifies to √ √ √ √ (3 + 2 )(1 + 2)2n−2 + (3 − 2 )(1 − 2)2n−2 Using P(n − 1), the above simplifies to √ √ 12N + 2a = 2(6 N + 2a), an...