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Omni-MATH / In Mathcity, there are infinitely many buses and infinitely many stations. The…

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problem

In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of 2:1,2,4,8,16,...2: 1, 2, 4, 8, 16, ... Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference rr and whose first term is the favourite number of the mayor. For which positive integers rr is it always possible that, no matter the favourite number of the mayor, given any mm stations, there is a bus going by all of them?
Plain-text mathematical notation (without MathML)
In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of 2:1,2,4,8,16,... Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference r and whose first term is the favourite number of the mayor. For which positive integers r is it always possible that, no matter the favourite number of the mayor, given any m stations, there is a bus going by all of them?
Original LaTeX notation
In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of $2: 1, 2, 4, 8, 16, ...$ Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference $r$ and whose first term is the favourite number of the mayor. For which positive integers $r$ is it always possible that, no matter the favourite number of the mayor, given any $m$ stations, there is a bus going by all of them?

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Source and history

task_id
c7145f5c-a64f-5e7d-9c15-4728de536ee8
task_key
test--c7145f5c-a64f-5e7d-9c15-4728de536ee8
task_revision_id
1
config / split
/ test

Official source

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