MathDB
Problems
Contests
International Contests
Benelux
2011 Benelux
2011 Benelux
Part of
Benelux
Subcontests
(4)
4
1
Hide problems
Writing numbers on a blackboard...
Abby and Brian play the following game: They first choose a positive integer
N
N
N
. Then they write numbers on a blackboard in turn. Abby starts by writing a
1
1
1
. Thereafter, when one of them has written the number
n
n
n
, the other writes down either
n
+
1
n + 1
n
+
1
or
2
n
2n
2
n
, provided that the number is not greater than
N
N
N
. The player who writes
N
N
N
on the blackboard wins. (a) Determine which player has a winning strategy if
N
=
2011
N = 2011
N
=
2011
. (b) Find the number of positive integers
N
⩽
2011
N\leqslant2011
N
⩽
2011
for which Brian has a winning strategy.(This is based on ISL 2004, Problem C5.)
3
1
Hide problems
Bounded Sequence and Cubes
If
k
k
k
is an integer, let
c
(
k
)
\mathrm{c}(k)
c
(
k
)
denote the largest cube that is less than or equal to
k
k
k
. Find all positive integers
p
p
p
for which the following sequence is bounded:
a
0
=
p
a_0 = p
a
0
=
p
and
a
n
+
1
=
3
a
n
−
2
c
(
a
n
)
a_{n+1} = 3a_n-2\mathrm{c}(a_n)
a
n
+
1
=
3
a
n
−
2
c
(
a
n
)
for
n
⩾
0
n \geqslant 0
n
⩾
0
.
2
1
Hide problems
Perpendicular Bisector of Angle Bisector, Concyclic Points
Let
A
B
C
ABC
A
BC
be a triangle with incentre
I
I
I
. The angle bisectors
A
I
AI
A
I
,
B
I
BI
B
I
and
C
I
CI
C
I
meet
[
B
C
]
[BC]
[
BC
]
,
[
C
A
]
[CA]
[
C
A
]
and
[
A
B
]
[AB]
[
A
B
]
at
D
D
D
,
E
E
E
and
F
F
F
, respectively. The perpendicular bisector of
[
A
D
]
[AD]
[
A
D
]
intersects the lines
B
I
BI
B
I
and
C
I
CI
C
I
at
M
M
M
and
N
N
N
, respectively. Show that
A
A
A
,
I
I
I
,
M
M
M
and
N
N
N
lie on a circle.
1
1
Hide problems
Benelux Couples
An ordered pair of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
with
1
<
m
<
n
1<m<n
1
<
m
<
n
is said to be a Benelux couple if the following two conditions hold:
m
m
m
has the same prime divisors as
n
n
n
, and
m
+
1
m+1
m
+
1
has the same prime divisors as
n
+
1
n+1
n
+
1
. (a) Find three Benelux couples
(
m
,
n
)
(m,n)
(
m
,
n
)
with
m
⩽
14
m\leqslant 14
m
⩽
14
. (b) Prove that there are infinitely many Benelux couples