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Close sum of digits

Source: Ukrainian Mathematical Olympiad 2021. Day 2, Problem 8.8

December 21, 2023
number theorysum of digits

Problem Statement

You are given kk positive integers a1,a2,…,aka_1, a_2, \ldots, a_k. Prove that there exists a positive integer NN, such that sums of digits of numbers Na1,Na2,…,NakNa_1, Na_2, \ldots, Na_k differ in less than 20212020\frac{2021}{2020} times. That is, if SmaxS_{max} is the largest sum of digits of any of these numbers, and SminS_{min} is the smallest, then SmaxSmin<20212020\frac{S_{max}}{S_{min}} < \frac{2021}{2020}
Proposed by Arsenii Nikolaiev
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