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National and Regional Contests
Turkey Contests
National Olympiad First Round
2013 National Olympiad First Round
28
28
Part of
2013 National Olympiad First Round
Problems
(1)
P28 [Combinatorics] - Turkish NMO 1st Round - 2013
Source:
4/20/2013
In the beginning, there is a pair of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
from the pairs
(
7
,
79
)
(7,79)
(
7
,
79
)
,
(
17
,
71
)
(17,71)
(
17
,
71
)
,
(
10
,
101
)
(10,101)
(
10
,
101
)
,
(
21
,
251
)
(21,251)
(
21
,
251
)
,
(
50
,
405
)
(50,405)
(
50
,
405
)
, can Alice guarantee to win when she makes the first move?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of above
<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
ceiling function