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Make almost equal

Source: Kyiv City MO 2021 Round 1, Problem 9.2

December 21, 2023
sum of digitsnumber theory

Problem Statement

Roma wrote on the board each of the numbers 2018,2019,20202018, 2019, 2020, 100100 times each. Let us denote by S(n)S(n) the sum of digits of positive integer nn. In one action, Roma can choose any positive integer kk and instead of any three numbers a,b,ca, b, c written on the board write the numbers 2S(a+b)+k,2S(b+c)+k2S(a + b) + k, 2S(b + c) + k and 2S(c+a)+k2S(c + a) + k. Can Roma after several such actions make 299299 numbers on the board equal, and the last one differing from them by 11?
Proposed by Oleksii Masalitin
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