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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1989 French Mathematical Olympiad
Problem 1
Problem 1
Part of
1989 French Mathematical Olympiad
Problems
(1)
central symmetries of plane figures
Source: France 1989 P1
5/18/2021
Given a figure
B
B
B
in the plane, consider the figures
A
A
A
, containing
B
\mathcal B
B
, with the property
(
P
)
(P)
(
P
)
: a composition of an odd number of central symmetries with centers in
A
A
A
is also a central symmetry with center in
A
A
A
. The task of this problem is to determine the smallest such figure, denoted by
A
\mathcal A
A
, that is contained in every figure
A
A
A
.(a) Determine the figure
A
\mathcal A
A
if
B
B
B
consists of:
(
1
)
(1)
(
1
)
two distinct points
I
,
J
I,J
I
,
J
;
(
2
)
(2)
(
2
)
three non-collinear points
I
,
J
,
K
I,J,K
I
,
J
,
K
. (b) Determine
A
\mathcal A
A
if
B
B
B
is a circle (with nonzero radius). (c) Give some examples of figures
B
B
B
whose associated figures
A
\mathcal A
A
are mutually distinct and distinct from the above ones.
geometry