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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2016 Iran MO (2nd Round)
4
4
Part of
2016 Iran MO (2nd Round)
Problems
(1)
N lines cutting each other in the plane
Source: Iranian Math Olympiad(Second Round 2016)
5/5/2016
Let
l
1
,
l
2
,
l
3
,
.
.
.
,
L
n
l_1,l_2,l_3,...,L_n
l
1
,
l
2
,
l
3
,
...
,
L
n
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let
A
A
A
be the intersection point of lines
l
i
,
l
j
l_i,l_j
l
i
,
l
j
. We call
A
A
A
an "Interior Point" if there are points
C
,
D
C,D
C
,
D
on
l
i
l_i
l
i
and
E
,
F
E,F
E
,
F
on
l
j
l_j
l
j
such that
A
A
A
is between
C
,
D
C,D
C
,
D
and
E
,
F
E,F
E
,
F
. Prove that there are at least
(
n
−
2
)
(
n
−
3
)
2
\frac{(n-2)(n-3)}{2}
2
(
n
−
2
)
(
n
−
3
)
Interior points.(
n
>
2
n>2
n
>
2
) note: by point here we mean the points which are intersection point of two of
l
1
,
l
2
,
.
.
.
,
l
n
l_1,l_2,...,l_n
l
1
,
l
2
,
...
,
l
n
.
combinatorics