Maths Olympiad Prep

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Combinatorics Difficulty 5.5 AIME, harder Prove it Saudi Arabia

Let nn be a positive integer. There are 2n2n knights sitting at a round table. They consist of nn pair of partners, each pair of which wishes to shake hands. A pair can shake hands only with next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight can meet her partner and shake hands at some moment.

Solution

Solution. (Solution of Ahmad Alshehri, IMO 2025's team member)
SAUDI ARABIAN IMO Booklet 2025
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## Saudi Booklet 2025 — Page 46
46
Solution of IMO Team selection tests
نلاحظ في المثال : ① ② ③ ④

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.