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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2000 National Olympiad First Round
23
23
Part of
2000 National Olympiad First Round
Problems
(1)
Turkey NMO 2000 1st Round - P23 (Combinatorics)
Source:
7/25/2012
A committee with
20
20
20
members votes for the candidates
A
,
B
,
C
A,B,C
A
,
B
,
C
by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes
B
A
C
BAC
B
A
C
, he prefers
B
B
B
to
A
A
A
and
C
C
C
, and prefers
A
A
A
to
C
C
C
). After the ballots are counted, it is recognized that each of the six different permutations of three candidates appears in at least one ballot, and
11
11
11
members prefer
A
A
A
to
B
B
B
,
12
12
12
members prefer
C
C
C
to
A
A
A
,
14
14
14
members prefer
B
B
B
to
C
C
C
. How many members are there such that
B
B
B
is the first choice of them?
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A
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5
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B
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7
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C
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8
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(
D
)
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10
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(
E
)
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More information is needed
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ \text{More information is needed}
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(
A
)
<
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5
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(
B
)
<
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7
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>
(
C
)
<
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>
8
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−
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>
(
D
)
<
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>
10
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−
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>
(
E
)
<
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>
More information is needed