IMO 1974 LL SWE33

Let a be a real number such that 0 < a < 1, and let n be a

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IMO 1974 LL SWE33

Origin: SWE

Problem

Let a be a real number such that 0 < a < 1, and let n be a positive integer. Define the sequence a0, a1, a2, . . . , an recursively by a0 = a; ak+1 = ak + 1 na2 k for k = 0, 1, . . ., n −1. Prove that there exists a real number A, depending on a but independent of n, such that 0 < n(A −an) < A3.