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KoMaL A Problems
KoMaL A Problems 2023/2024
A. 868
A. 868
Part of
KoMaL A Problems 2023/2024
Problems
(1)
Collineation destroying ratios?
Source: KoMaL A. 868
1/13/2024
A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between
100
/
101
100/101
100/101
and
101
/
100
101/100
101/100
, or at least
100
100
100
or at most
1
/
100
1/100
1/100
. Is it true that for any distinct points
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots,A_n
A
1
,
A
2
,
…
,
A
n
in the plane it is always possible to find distinct points
A
1
′
,
A
2
′
,
…
,
A
n
′
A_1',A_2',\ldots, A_n'
A
1
′
,
A
2
′
,
…
,
A
n
′
that form a disharmonic set of points, and moreover
A
i
,
A
j
A_i, A_j
A
i
,
A
j
and
A
k
A_k
A
k
are collinear in this order if and only if
A
i
′
,
A
j
′
A_i', A_j'
A
i
′
,
A
j
′
and
A
k
′
A_k'
A
k
′
are collinear in this order (for all distinct
1
≤
i
,
j
,
k
≤
n
1 \le i,j,k\le n
1
≤
i
,
j
,
k
≤
n
?Submitted by Dömötör Pálvölgyi and Balázs Keszegh, Budapest
geometry