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National and Regional Contests
Korea Contests
Korea National Olympiad
2013 Korea National Olympiad
8
8
Part of
2013 Korea National Olympiad
Problems
(1)
Two lines separating plane
Source: Korea National 2013 #8
11/10/2013
For positive integer
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
there are
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
points on plane which none of three are collinear. Prove there exist two lines
l
1
,
l
2
l_1, l_2
l
1
,
l
2
such that (1)
l
1
,
l
2
l_1, l_2
l
1
,
l
2
are not parallel. (2)
l
1
,
l
2
l_1, l_2
l
1
,
l
2
do not pass through any of
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
points. (3) There are
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
points on each region separated by two lines
l
1
,
l
2
l_1, l_2
l
1
,
l
2
.
combinatorics
Korea