SIMON-MARAIS 2024 A3

Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a, b)$ be the largest integer satisfying $m(a, b) \leq \frac{1 + na}{1 + nb}$ for all integers $n \geq 1$.

(a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that

$$m(a, b) \leq \frac{1 + na}{W + nb},$$

for all $n \geq k$.

(b) For each $(a, b) \in S$, let $k(a, b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max\{k(a, b) \mid (a, b) \in S\}$ or prove that it does not exist.