MathDB

We are given

100

strictly increasing sequences of positive integers:

A_{i}= (a_{1}^{(i)}, a_{2}^{(i)},...), i = 1, 2,..., 100

. For

1 \leq r, s \leq 100

we deﬁne the following quantities:

f_{r}(u)=

the number of elements of

A_{r}

not exceeding

n

;

f_{r,s}(u) =

the number of elements of

A_{r}\cap A_{s}

not exceeding

n

. Suppose that

f_{r}(n) \geq\frac{1}{2}n

for all

r

and

n

. Prove that there exists a pair of indices

(r, s)

with

r \not = s

such that

f_{r,s}(n) \geq\frac{8n}{33}

for at least ﬁve distinct

n-s

with

1 \leq n < 19920.

Problem Statement