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Omni-MATH / Let n≥3 be a fixed integer. The number 1 is written n times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of…

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problem

Let n≥3n \geq 3 be a fixed integer. The number 1 is written nn times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers aa and bb, replacing them with the numbers 1 and a+ba+b, then adding one stone to the first bucket and gcd(a,b)\operatorname{gcd}(a, b) stones to the second bucket. After some finite number of moves, there are ss stones in the first bucket and tt stones in the second bucket, where ss and tt are positive integers. Find all possible values of the ratio ts\frac{t}{s}.
Plain-text mathematical notation (without MathML)
Let n≥3 be a fixed integer. The number 1 is written n times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers a and b, replacing them with the numbers 1 and a+b, then adding one stone to the first bucket and gcd(a,b) stones to the second bucket. After some finite number of moves, there are s stones in the first bucket and t stones in the second bucket, where s and t are positive integers. Find all possible values of the ratio (t)/(s).
Original LaTeX notation
Let \(n \geq 3\) be a fixed integer. The number 1 is written \(n\) times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers \(a\) and \(b\), replacing them with the numbers 1 and \(a+b\), then adding one stone to the first bucket and \(\operatorname{gcd}(a, b)\) stones to the second bucket. After some finite number of moves, there are \(s\) stones in the first bucket and \(t\) stones in the second bucket, where \(s\) and \(t\) are positive integers. Find all possible values of the ratio \(\frac{t}{s}\).

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Source and history

task_id
65282ae9-5b1f-5fbe-8a48-19a4862b095c
task_key
test--65282ae9-5b1f-5fbe-8a48-19a4862b095c
task_revision_id
1
config / split
/ test

Official source

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