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National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2015 JBMO TST - Turkey
8
8
Part of
2015 JBMO TST - Turkey
Problems
(1)
"Critical" Coloring of 100^2 Points
Source: Turkey JBMO TST 2015 P8
6/23/2016
A coloring of all plane points with coordinates belonging to the set
S
=
{
0
,
1
,
…
,
99
}
S=\{0,1,\ldots,99\}
S
=
{
0
,
1
,
…
,
99
}
into red and white colors is said to be critical if for each
i
,
j
∈
S
i,j\in S
i
,
j
∈
S
at least one of the four points
(
i
,
j
)
,
(
i
+
1
,
j
)
,
(
i
,
j
+
1
)
(i,j),(i + 1,j),(i,j + 1)
(
i
,
j
)
,
(
i
+
1
,
j
)
,
(
i
,
j
+
1
)
and
(
i
+
1
,
j
+
1
)
(i + 1, j + 1)
(
i
+
1
,
j
+
1
)
(
99
+
1
≡
0
)
(99 + 1\equiv0)
(
99
+
1
≡
0
)
is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
combinatorics