Home > Algebra > IMC 2014 Day 2 Problem 1

IMC 2014 Day 2 Problem 1


For a positive integer {x} denote its {n}-th decimal digit by {d_n(x)}, i.e. {d_n(x) \in \{0,1,..,9\}} and {x = \displaystyle \sum_{n=1}^\infty d_n(x) 10^{n-1}}. Suppose that for some sequence {(a_n)_{n=1}^\infty} there are only finitely many zeros in the sequence {(d_n(a_n))_{n=1}^\infty}. Prove that there are infinitely many positive integers that do not occur in the sequence {(a_n)_{n=1}^\infty}.

IMC 2014 Day 2 Problem 1

Hint: Since d_n(a_n)\neq 0 it follows that a_n > 10^{n-1} (for n large enough).

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