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National Olympiad First Round
2012 National Olympiad First Round
16
16
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 16 {Combinatorics}
Source:
7/1/2012
Every cell of
8
×
8
8\times8
8
×
8
chessboard contains either
1
1
1
or
−
1
-1
−
1
. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than
−
3
-3
−
3
?
<
s
p
a
n
c
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a
s
s
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′
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a
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x
−
b
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>
(
A
)
<
/
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a
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>
6
<
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>
(
B
)
<
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5
<
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a
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−
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>
(
C
)
<
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4
<
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a
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−
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>
(
D
)
<
/
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>
3
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a
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x
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>
(
E
)
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2
<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 2
<
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c
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(
A
)
<
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6
<
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c
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a
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=
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x
−
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>
(
B
)
<
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>
5
<
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=
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x
−
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>
(
C
)
<
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4
<
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′
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x
−
b
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′
>
(
D
)
<
/
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p
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>
3
<
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p
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l
a
ss
=
′
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a
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x
−
b
o
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d
′
>
(
E
)
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2