MathDB
VMEO IV October P4

Source:

December 26, 2016
combinatorics

Problem Statement

Let n∈Z+n\in\mathbb{Z}^+. Arrange nn students A1,A2,...,AnA_1,A_2,...,A_n on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is m≥nm\geq n. A configuration is called balance if for an arbitrary student AiA_i, there will always be a regular polygon taking AiA_i as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies.
a) Given nn, find the least mm such that we can create a balance configuration.
b) In a move, a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if nn is the product of at most 22 prime numbers and mm satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.
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