2022 IMO Problems/Problem 1

Problem

The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k\le 2n$ , Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k^{th}$ coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n = 4$ and $k = 4$ , the process starting from the ordering AABBBABA would be

AABBBABA → BBBAAABA → AAABBBBA → BBBBAAAA → BBBBAAAA → ...

Find all pairs $(n, k)$ with $1 \le k \le 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

Solution

https://www.youtube.com/watch?v=nYD-qIOdi_c [Video contains solutions to all day 1 problems] https://youtu.be/KHn3xD6wS3A [Video contains problem 1 discussion]

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2022 IMO Problems/Problem 1

Problem

Solution