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Problems
Contests
International Contests
Benelux
2017 Benelux
2017 Benelux
Part of
Benelux
Subcontests
(4)
4
1
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Benelux n-square
A Benelux n-square (with
n
≥
2
n\geq 2
n
≥
2
) is an
n
×
n
n\times n
n
×
n
grid consisting of
n
2
n^2
n
2
cells, each of them containing a positive integer, satisfying the following conditions:
∙
\bullet
∙
the
n
2
n^2
n
2
positive integers are pairwise distinct.
∙
\bullet
∙
if for each row and each column we compute the greatest common divisor of the
n
n
n
numbers in that row/column, then we obtain
2
n
2n
2
n
different outcomes.(a) Prove that, in each Benelux n-square (with
n
≥
2
n \geq 2
n
≥
2
), there exists a cell containing a number which is at least
2
n
2
.
2n^2.
2
n
2
.
(b) Call a Benelux n-square minimal if all
n
2
n^2
n
2
numbers in the cells are at most
2
n
2
.
2n^2.
2
n
2
.
Determine all
n
≥
2
n\geq 2
n
≥
2
for which there exists a minimal Benelux n-square.
3
1
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Another perpendicularity
In the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
we have
∠
B
=
∠
C
\angle B = \angle C
∠
B
=
∠
C
and
∠
D
=
9
0
∘
.
\angle D = 90^{\circ}.
∠
D
=
9
0
∘
.
Suppose that
∣
A
B
∣
=
2
∣
C
D
∣
.
|AB| = 2|CD|.
∣
A
B
∣
=
2∣
C
D
∣.
Prove that the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
is perpendicular to
C
D
.
CD.
C
D
.
2
1
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Alice and Bob play a game...
Let
n
≥
2
n\geq 2
n
≥
2
be an integer. Alice and Bob play a game concerning a country made of
n
n
n
islands. Exactly two of those
n
n
n
islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands
I
1
I_1
I
1
and
I
2
I_2
I
2
such that:
∙
\bullet
∙
I
1
I_1
I
1
and
I
2
I_2
I
2
are not already connected by a bridge.
∙
\bullet
∙
at least one of the two islands
I
1
I_1
I
1
and
I
2
I_2
I
2
is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.) As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each
n
≥
2
n\geq 2
n
≥
2
) who has a winning strategy. (Note: It is allowed to construct a bridge passing above another bridge.)
1
1
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Functional equation
Find all functions
f
:
Q
>
0
→
Z
>
0
f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}
f
:
Q
>
0
→
Z
>
0
such that
f
(
x
y
)
⋅
gcd
(
f
(
x
)
f
(
y
)
,
f
(
1
x
)
f
(
1
y
)
)
=
x
y
f
(
1
x
)
f
(
1
y
)
,
f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right) = xyf(\frac{1}{x})f(\frac{1}{y}),
f
(
x
y
)
⋅
g
cd
(
f
(
x
)
f
(
y
)
,
f
(
x
1
)
f
(
y
1
)
)
=
x
y
f
(
x
1
)
f
(
y
1
)
,
for all
x
,
y
∈
Q
>
0
,
x, y \in \Bbb{Q}_{>0,}
x
,
y
∈
Q
>
0
,
where
gcd
(
a
,
b
)
\gcd(a, b)
g
cd
(
a
,
b
)
denotes the greatest common divisor of
a
a
a
and
b
.
b.
b
.