MathDB
Maximum number of m-tastic numbers

Source: IMO Shortlist 2017 N4

July 10, 2018
IMO Shortlistnumber theorydecimal representationDigitsIMOIMO 2017

Problem Statement

Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer mm, we say that a positive integer tt is m−m-tastic if there exists a number c∈{1,2,3,…,2017}c\in \{1,2,3,\ldots ,2017\} such that 10t−1c⋅m\dfrac{10^t-1}{c\cdot m} is short, and such that 10k−1c⋅m\dfrac{10^k-1}{c\cdot m} is not short for any 1≤k<t1\le k<t. Let S(m)S(m) be the set of m−m-tastic numbers. Consider S(m)S(m) for m=1,2,….m=1,2,\ldots{}. What is the maximum number of elements in S(m)S(m)?
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