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Greece Junior Math Olympiad
2010 Greece Junior Math Olympiad
4
4
Part of
2010 Greece Junior Math Olympiad
Problems
(1)
Combinatorics
Source: Archimedes Junior 2010
3/17/2020
Three parallel lines
ℓ
1
,
ℓ
2
\ell_1, \ell_2
ℓ
1
,
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
of a plane are given such that the line
ℓ
2
\ell_2
ℓ
2
has the same distance
a
a
a
from
ℓ
1
\ell_1
ℓ
1
and
ℓ
3
\ell_3
ℓ
3
. We put
5
5
5
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
5
M_5
M
5
on the lines
ℓ
1
,
ℓ
2
\ell_1, \ell_2
ℓ
1
,
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
in such a way that each line contains at least one point. Detennine the maximal number of isosceles triangles that are possible to be formed with vertices three of the 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
5
M_5
M
5
in the following cases: (i)
M
1
,
M
2
,
M
3
∈
ℓ
2
,
M
4
∈
ℓ
1
M_1,M_2,M_3 \in \ell_2, M_4 \in \ell_1
M
1
,
M
2
,
M
3
∈
ℓ
2
,
M
4
∈
ℓ
1
and
M
5
∈
ℓ
3
M_5 \in \ell_3
M
5
∈
ℓ
3
. (ii)
M
1
,
M
2
∈
ℓ
1
,
M
3
,
M
4
∈
ℓ
3
M_1,M_2 \in \ell_1, M_3,M_4 \in \ell_3
M
1
,
M
2
∈
ℓ
1
,
M
3
,
M
4
∈
ℓ
3
and
M
5
∈
ℓ
2
M_5 \in \ell_2
M
5
∈
ℓ
2
.
combinatorics