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monving balls in 2018 boxes

Source: 1st Mathematics Regional Olympiad of Mexico Northwest 2018 P1

September 6, 2022
combinatorics

Problem Statement

There are 20182018 boxes C1C_1, C2C_2, C3C_3,..,C2018C_{2018}. The nn-th box CnC_n contains nn balls. A move consists of the following steps: a) Choose an integer kk greater than 11 and choose mm a multiple of kk. b) Take a ball from each of the consecutive boxes Cm1C_{m-1}, CmC_m, Cm+1C_{m+1} and move the 33 balls to the box Cm+kC_{m+k}. With these movements, what is the largest number of balls we can get in the box 20182018?