SIMON-MARAIS 2022 B4

The following problem is open in the sense that the answer to part (b) is not currently known. A solution to part (a) will be awarded 7 points. Up to 7 additional points may be awarded for progress on part (b).

Let $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ be two sequences of positive integers, satisfying $a_0b_0 \geq 600$ and

$$a_{n+1} = a_n + 2 \cdot \lfloor b_n/20 \rfloor,$$

$$b_{n+1} = b_n + 3 \cdot \lfloor a_n/30 \rfloor,$$

for all $n \geq 0$.

(a) Prove that there exists a nonnegative integer $N$ such that

$$-13 \leq a_n - b_n \leq 23,$$

for all $n \geq N$.

(b) Must there exist a nonnegative integer $n$ such that $a_n = b_n$?

Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.