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Source: Mongolian Mathematical Olympiad P6

January 30, 2024
number theorySequences

Problem Statement

Let mm be a positive integer. We say that a sequence of positive integers written on a circle is good , if the sum of any mm consecutive numbers on this circle is a power of mm.
1. Let n≥2n \geq 2 be a positive integer. Prove that for any good sequence with mnmn numbers, we can remove mm numbers such that the remaining mn−mmn-m numbers form a good sequence.
2. Prove that in any good sequence with m2m^2 numbers, we can always find a number that was repeated at least mm times in the sequence.
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