MathDB
n+i stones at P_i points on a circle

Source: 2020 Colombia day 2 p6

November 27, 2022
combinatorics

Problem Statement

Let kk be a positive integer and n1,n2,...,nkn_1, n_2, ..., n_k be non-negative integers. Points P1,P2,...,PkP_1, P_2, ..., P_k lie on a circle in such a way that at point PiP_i there are nin_i stones. Leandro wishes to change the position of some of these stones in order to accomplish his objective which is to have the same number of stones at each point of the circle. He does this by repeating as many times as necessary the following operation: if there exists a point on the circle with at least k−1k - 1 stones, he can choose k−1k -1 of these and distribute them by giving one to each of the remaining k−1k - 1 points. For which values n1,n2,...,nkn_1, n_2, ..., n_k can Leandro accomplish his objective?
In the figure below there is a configuration of stones for k=4k = 4. On the right is the initial division of stones, while on the left there is the configuration obtained from the initial one by choosing k−1=3k - 1 = 3 stones from the top point on the circle and distributing one each to the other points.
[figures missing]
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