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National and Regional Contests
France Contests
French Mathematical Olympiad
1998 French Mathematical Olympiad
Problem 4
Problem 4
Part of
1998 French Mathematical Olympiad
Problems
(1)
intersection of two lines and existence of points
Source: 1998 France MO P4
4/10/2021
Let there be given two lines
D
1
D_1
D
1
and
D
2
D_2
D
2
which intersect at point
O
O
O
, and a point
M
M
M
not on any of these lines. Consider two variable points
A
∈
D
1
A\in D_1
A
∈
D
1
and
b
∈
D
2
b\in D_2
b
∈
D
2
such that
M
M
M
belongs to the segment
A
B
AB
A
B
.(a) Prove that there exists a position of
A
A
A
and
B
B
B
for which the area of triangle
O
A
B
OAB
O
A
B
is minimal. Construct such points
A
A
A
and
B
B
B
. (b) Prove that there exists a position of
A
A
A
and
B
B
B
for which the area of triangle
O
A
B
OAB
O
A
B
is minimal. Show that for such
A
A
A
and
B
B
B
, the perimeters of
△
O
A
M
\triangle OAM
△
O
A
M
and
△
O
B
M
\triangle OBM
△
OBM
are equal, and that
A
M
tan
1
2
∠
O
A
M
=
B
M
tan
1
2
∠
O
B
M
\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}
t
a
n
2
1
∠
O
A
M
A
M
=
t
a
n
2
1
∠
OBM
BM
. Construct such points
A
A
A
and
B
B
B
.
geometry