MathDB

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.

Problem Statement