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Problems
Contests
International Contests
Nordic
2010 Nordic
2010 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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Representation as a sum of simple positive integers
A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer
k
k
k
such that each positive integer
n
n
n
can be written as
n
=
a
1
±
a
2
±
a
3
±
⋯
±
a
k
n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k
n
=
a
1
±
a
2
±
a
3
±
⋯
±
a
k
where
a
1
,
…
,
a
k
a_1, \dots , a_k
a
1
,
…
,
a
k
are simple.
3
1
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2010 buttons connected to 2010 lamps
Laura has
2010
2010
2010
lamps connected with
2010
2010
2010
buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?
2
1
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Three circles with a common point of intersection
Three circles
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
and
Γ
C
\Gamma_C
Γ
C
share a common point of intersection
O
O
O
. The other common point of
Γ
A
\Gamma_A
Γ
A
and
Γ
B
\Gamma_B
Γ
B
is
C
C
C
, that of
Γ
A
\Gamma_A
Γ
A
and
Γ
C
\Gamma_C
Γ
C
is
B
B
B
, and that of
Γ
C
\Gamma_C
Γ
C
and
Γ
B
\Gamma_B
Γ
B
is
A
A
A
. The line
A
O
AO
A
O
intersects the circle
Γ
A
\Gamma_A
Γ
A
in the point
X
≠
O
X \ne O
X
=
O
. Similarly, the line
B
O
BO
BO
intersects the circle
Γ
B
\Gamma_B
Γ
B
in the point
Y
≠
O
Y \ne O
Y
=
O
, and the line
C
O
CO
CO
intersects the circle
Γ
C
\Gamma_C
Γ
C
in the point
Z
≠
O
Z \ne O
Z
=
O
. Show that
∣
A
Y
∣
⋅
∣
B
Z
∣
⋅
∣
C
X
∣
∣
A
Z
∣
⋅
∣
B
X
∣
⋅
∣
C
Y
∣
=
1.
\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.
∣
A
Z
∣
⋅
∣
BX
∣
⋅
∣
C
Y
∣
∣
A
Y
∣
⋅
∣
BZ
∣
⋅
∣
CX
∣
=
1.
1
1
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Inequality of a non-decreasing multiplicative function
A function
f
:
Z
+
→
Z
+
f : \mathbb{Z}_+ \to \mathbb{Z}_+
f
:
Z
+
→
Z
+
, where
Z
+
\mathbb{Z}_+
Z
+
is the set of positive integers, is non-decreasing and satisfies
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn) = f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
for all relatively prime positive integers
m
m
m
and
n
n
n
. Prove that
f
(
8
)
f
(
13
)
≥
(
f
(
10
)
)
2
f(8)f(13) \ge (f(10))^2
f
(
8
)
f
(
13
)
≥
(
f
(
10
)
)
2
.