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National and Regional Contests
Turkey Contests
National Olympiad First Round
2000 National Olympiad First Round
15
15
Part of
2000 National Olympiad First Round
Problems
(1)
Turkey NMO 2000 1st Round - P15 (Combinatorics)
Source:
7/25/2012
A
,
B
,
C
A,B,C
A
,
B
,
C
are playing backgammon tournament. At first,
A
A
A
plays with
B
B
B
. Then the winner plays with
C
C
C
. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that
C
C
C
wins the tournament?
<
s
p
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a
s
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x
−
b
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>
(
A
)
<
/
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>
2
7
<
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(
B
)
<
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1
3
<
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>
(
C
)
<
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>
3
14
<
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c
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a
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=
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x
−
b
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>
(
D
)
<
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1
7
<
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c
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a
s
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=
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x
−
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>
(
E
)
<
/
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p
a
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>
None
<span class='latex-bold'>(A)</span>\ \frac27 \qquad<span class='latex-bold'>(B)</span>\ \frac13 \qquad<span class='latex-bold'>(C)</span>\ \frac3{14} \qquad<span class='latex-bold'>(D)</span>\ \frac 17 \qquad<span class='latex-bold'>(E)</span>\ \text{None}
<
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an
c
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>
(
A
)
<
/
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p
an
>
7
2
<
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>
(
B
)
<
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>
3
1
<
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=
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a
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x
−
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>
(
C
)
<
/
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p
an
>
14
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
7
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
None
probability