IMO 1989 LL BUL5

Origin: BUL

Problem

The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal- ities a0 = \sqrt 2 , an+1 = \sqrt 1 − 1 −a2n, n = 0, 1, 2, . . . and b0 = 1, bn+1 = 1 + b2n −1 bn , n = 0, 1, 2, . . .. Prove the inequalities 2n+2an < \pi < 2n+2bn, for every n = 0, 1, 2, . . . .