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Math Prize 2013 Problem 20

Source:

September 10, 2013
inequalitiestrigonometry

Problem Statement

Let a0a_0, a1a_1, a2a_2, …\dots be an infinite sequence of real numbers such that a0=45a_0 = \frac{4}{5} and an=2an−12−1 a_{n} = 2 a_{n-1}^2 - 1 for every positive integer nn. Let cc be the smallest number such that for every positive integer nn, the product of the first nn terms satisfies the inequality a0a1…an−1≤c2n. a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. What is the value of 100c100c, rounded to the nearest integer?
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