MathDB

A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between

100/101

and

101/100

, or at least

100

or at most

1/100

. Is it true that for any distinct points

A_1,A_2,\ldots,A_n

in the plane it is always possible to find distinct points

A_1',A_2',\ldots, A_n'

that form a disharmonic set of points, and moreover

A_i, A_j

and

A_k

are collinear in this order if and only if

A_i', A_j'

and

A_k'

are collinear in this order (for all distinct

1 \le i,j,k\le n

?
Submitted by Dömötör Pálvölgyi and Balázs Keszegh, Budapest

Problem Statement