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2012 Greece Junior Math Olympiad
4
4
Part of
2012 Greece Junior Math Olympiad
Problems
(1)
Combinatorics
Source: Archimedes Junior 2012
3/16/2020
On a plane
Π
\Pi
Π
is given a straight line
ℓ
\ell
ℓ
and on the line
ℓ
\ell
ℓ
are given two different points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
. We consider on the plane
Π
\Pi
Π
, outside the line
ℓ
\ell
ℓ
, two different points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
. Examine if it is possible to put points
A
3
A_3
A
3
and
A
4
A_4
A
4
on such positions such the four points
A
1
,
A
2
,
A
3
,
A
4
A_1, A_2, A_3, A_4
A
1
,
A
2
,
A
3
,
A
4
form the maximal number of possible isosceles triangles, in the following cases: (a) when the points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
belong to dierent semi-planes with respect to
ℓ
\ell
ℓ
; (b) when the points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
belong to the same semi-planes with respect to
ℓ
\ell
ℓ
. Give all possible cases and explain how is possible to construct in each case the points
A
3
A_3
A
3
and
A
4
A_4
A
4
.
combinatorics