MathDB

Let

S

be a set not empty of positive integers and

AB

a segment with, initially, only points

A

and

B

colored by red. An operation consists of choosing two distinct points

X, Y

colored already by red and

n\in S

an integer, and painting in red the

n

points

A_1, A_2,..., A_n

of segment

XY

such that

XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY

and

XA_1<XA_2<...<XA_n

. Find the least positive integer

m

such exists a subset

S

\{1,2,.., m\}

such that, after a finite number of operations, we can paint in red the point

K

in the segment

AB

defined by

\frac{AK}{KB}= \frac{2709}{2022}

. Also, find the number of such subsets for such a value of

m

Problem Statement