Source: Mathcenter Contest / Oly - Thai Forum 2010 R1 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
Tags: number theory, Digits, power of 2
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Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$.
(tatari/nightmare)
Do you mean there exists some integer $n$ (rather than $n$ integers) such that the $k$th digit from the right of $2^n$ is $d$?
indeed, this makes sense, thanks for the suggestion, I do not speak Thai, so google translate does all the hard work
It was the Problem 3 of Korea National Olympiad 2021: https://artofproblemsolving.com/community/c6h2718205p23639576