Beaver Math Olympiad

Beaver Mathematical Olympiad (BMO) is an attempt to re-formulate the halting problem for some particular Turing machines as a mathematical problem in a style suitable for a hypothetical math olympiad.

The purpose of the BMO is twofold. First, statements where every non-essential details (e.g. related to tape encoding, number of steps, etc) are discarded are more suitable to be shared with mathematicians who perhaps are able to help. Second, it's a way to jokingly highlight how a hard question could appear deceptively simple.

Unsolved problems

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Let $(a_n{)}_{n\ge 1}$ and $(b_n{)}_{n\ge 1}$ be two sequences such that $(a_1,b_1)=(1,2)$ and

$$(a_{n+1},b_{n+1})=\{\begin{array}{ll}(a_n-b_n,4b_n+2)& \text{if }{a}_n\ge b_n\\ (2a_n+1,b_n-a_n)& \text{if }{a}_n<b_n\end{array}$$

for all positive integers $n$. Does there exist a positive integer $i$ such that $a_i=b_i$?

Hydra and Antihydra

Let $(a_n{)}_{n\ge 0}$ be a sequence such that $a_{n+1}=a_n+\lfloor \frac{a_n}{2}\rfloor$ for all non-negative integers $n$.

1. If $a_0=3$, does there exist a non-negative integer $k$ such that the list of numbers $a_0,a_1,a_2,\dots ,a_k$ have more than twice as many even numbers as odd numbers? (Hydra)
2. If $a_0=8$, does there exist a non-negative integer $k$ such that the list of numbers $a_0,a_1,a_2,\dots ,a_k$ have more than twice as many odd numbers as even numbers? (Antihydra)