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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
32
32
Part of
2014 National Olympiad First Round
Problems
(1)
P32 [Combinatorics] - Turkish NMO 1st Round - 2014
Source:
5/26/2014
There are
k
k
k
stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each
k
=
5
,
6
,
7
,
8
,
9
k=5,6,7,8,9
k
=
5
,
6
,
7
,
8
,
9
. If Alper is always the first player, for how many of the games can Alper guarantee that he does not lose the game?
<
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p
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c
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a
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>
(
A
)
<
/
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>
1
<
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=
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(
B
)
<
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2
<
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−
b
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>
(
C
)
<
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p
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>
3
<
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p
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c
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a
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s
=
′
l
a
t
e
x
−
b
o
l
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′
>
(
D
)
<
/
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p
a
n
>
4
<
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p
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c
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a
s
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=
′
l
a
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x
−
b
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>
(
E
)
<
/
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>
5
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5
<
s
p
an
c
l
a
ss
=
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a
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−
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>
(
A
)
<
/
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p
an
>
1
<
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p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
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′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
5