APMO 2020 P5

Discussion on Asian Pacific Mathematical Olympiad (APMO)
Soumitro_Shovon
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Joined:Fri Aug 21, 2020 11:39 am
APMO 2020 P5

Unread post by Soumitro_Shovon » Thu Dec 03, 2020 9:24 pm

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 $\gcd(a, b)$ 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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