MathDB
open - closed numbered doors in corridor

Source: 9th QEDMO p9 Seniors (21-22. 9. 2011) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 15, 2021
combinatoricsnumber theory

Problem Statement

In a very long corridor there is an infinite number of cabinets, which start with 1,2,3,...1,2,3,... numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set A⊆{1,2,3,...}A \subseteq \{1, 2,3,...\} . In ascending order, the QED people now cause chaos: the person with number a∈Aa \in A visits the cabinet with the numbers a,2a,3a,...a,2a,3a,... opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from AA open.
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