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France Contests
French Mathematical Olympiad
1989 French Mathematical Olympiad
Problem 2
Problem 2
Part of
1989 French Mathematical Olympiad
Problems
(1)
complex numbers form trapezoid (a), equality of locuses (b)
Source: France 1989 P2
5/18/2021
(a) Let
z
1
,
z
2
z_1,z_2
z
1
,
z
2
be complex numbers such that
z
1
z
2
=
1
z_1z_2=1
z
1
z
2
=
1
and
∣
z
1
−
z
2
∣
=
2
|z_1-z_2|=2
∣
z
1
−
z
2
∣
=
2
. Let
A
,
B
,
M
1
,
M
2
A,B,M_1,M_2
A
,
B
,
M
1
,
M
2
denote the points in complex plane corresponding to
−
1
,
1
,
z
1
,
z
2
-1,1,z_1,z_2
−
1
,
1
,
z
1
,
z
2
, respectively. Show that
A
M
1
B
M
2
AM_1BM_2
A
M
1
B
M
2
is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
be circles in the plane with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, and with radius
d
2
d\sqrt2
d
2
, where
2
d
=
O
1
O
2
2d=O_1O_2
2
d
=
O
1
O
2
. Let
P
P
P
and
Q
Q
Q
be two variable points on
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
respectively, both on
O
1
O
2
O_1O_2
O
1
O
2
on on different sides of
O
1
O
2
O_1O_2
O
1
O
2
, such that
P
Q
=
2
d
PQ=2d
PQ
=
2
d
. Prove that the locus of midpoints
I
I
I
of segments
P
Q
PQ
PQ
is the same as the locus of points
M
M
M
with
M
O
1
⋅
M
O
2
=
m
MO_1\cdot MO_2=m
M
O
1
⋅
M
O
2
=
m
for some
m
m
m
.
complex numbers
geometry