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 $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be two sequences such that $(a_{1},b_{1})=(1,2)$ and

$$(a_{n+1},b_{n+1})={\begin{cases}(a_{n}-b_{n},4b_{n}+2)&{\text{if }}a_{n}\geq b_{n}\\(2a_{n}+1,b_{n}-a_{n})&{\text{if }}a_{n}<b_{n}\end{cases}}$$

for all positive integers $n$. Does there exist a positive integer $i$ such that $a_{i}=b_{i}$?

Hydra and Antihydra

Let $(a_{n})_{n\geq 0}$ be a sequence such that $a_{n+1}=a_{n}+\left\lfloor {\frac {a_{n}}{2}}\right\rfloor$ 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)