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National and Regional Contests
France Contests
French Mathematical Olympiad
1988 French Mathematical Olympiad
Problem 4
Problem 4
Part of
1988 French Mathematical Olympiad
Problems
(1)
circle with 2n+1 points, product of distances
Source: France 1988 P4
5/18/2021
A circle
C
\mathcal C
C
and five distinct points
M
1
,
M
2
,
M
3
,
M
4
M_1,M_2,M_3,M_4
M
1
,
M
2
,
M
3
,
M
4
and
M
M
M
on
C
\mathcal C
C
are given in the plane. Prove that the product of the distances from
M
M
M
to lines
M
1
M
2
M_1M_2
M
1
M
2
and
M
3
M
4
M_3M_4
M
3
M
4
is equal to the product of the distances from
M
M
M
to the lines
M
1
M
3
M_1M_3
M
1
M
3
and
M
2
M
4
M_2M_4
M
2
M
4
. What can one deduce for
2
n
+
1
2n+1
2
n
+
1
distinct points
M
1
,
…
,
M
2
n
,
M
M_1,\ldots,M_{2n},M
M
1
,
…
,
M
2
n
,
M
on
C
\mathcal C
C
?
geometry