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Determine whether the number is rational

Source: Bulgarian IMO TST 2008, Day 2, Problem 1

July 8, 2013
number theory proposednumber theory

Problem Statement

For each positive integer nn, denote by ana_{n} the first digit of 2n2^{n} (base ten). Is the number 0.a1a2a30.a_{1}a_{2}a_{3}\cdots rational?