IMO Shortlist 2017 problem C7


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3. listopada 2019.
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For any finite sets X and Y of positive integers, denote by f_X(k) the k^{\text{th}} smallest positive integer not in X, and let X*Y=X\cup \{ f_X(y):y\in Y\}.Let A be a set of a>0 positive integers and let B be a set of b>0 positive integers. Prove that if A*B=B*A, then \underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.

Izvor: https://www.imo-official.org/problems/IMO2017SL.pdf