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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2008 Switzerland - Final Round
2008 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
6
1
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(p + q) / (p - q) is odd natural
Determine all odd natural numbers of the form
p
+
q
p
−
q
,
\frac{p + q}{p - q},
p
−
q
p
+
q
,
where
p
>
q
p > q
p
>
q
are prime numbers.
10
1
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x + y^2 + z^3 + w^6 >=a (xyzw)^b
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive real numbers with the following properties: (i) For all positive real numbers
x
,
y
,
z
,
w
x, y, z,w
x
,
y
,
z
,
w
holds
x
+
y
2
+
z
3
+
w
6
≥
a
(
x
y
z
w
)
b
x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}
x
+
y
2
+
z
3
+
w
6
≥
a
(
x
yz
w
)
b
. (ii) There is a quadruple
(
x
,
y
,
z
,
w
)
(x, y, z,w)
(
x
,
y
,
z
,
w
)
of positive real numbers such that in equality (i) applies.
7
1
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21 parts of 8x11 rectangle
An
8
×
11
8 \times 11
8
×
11
rectangle of unit squares somehow becomes disassembled into
21
21
21
contiguous parts . Prove that at least two of these parts, except for rotations and reflections have the same shape.
4
1
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3x1 papers cover n x n x n cube
Consider three sides of an
n
×
n
×
n
n \times n \times n
n
×
n
×
n
cube that meet at one of the corners of the cube. For which
n
n
n
is it possible to use this completely and without overlapping to cover strips of paper of size
3
×
1
3 \times 1
3
×
1
? The paper strips can also do this glued over the edges between these cube faces.
3
1
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2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}} divisible by 2008
Show that each number is of the form
2
5
2
5
.
.
.
+
4
5
4
5
.
.
.
2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}
2
5
2
5
...
+
4
5
4
5
...
is divisible by
2008
2008
2008
, where the exponential towers can be any independent ones have height
≥
3
\ge 3
≥
3
.
2
1
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f(xy) <= 1/2 (xf(y) + yf(x) )
Determine all functions
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
, so that for all
x
,
y
>
0
x, y > 0
x
,
y
>
0
:
f
(
x
y
)
≤
x
f
(
y
)
+
y
f
(
x
)
2
f(xy) \le \frac{xf(y) + yf(x)}{2}
f
(
x
y
)
≤
2
x
f
(
y
)
+
y
f
(
x
)
9
1
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7 lines in the plane and good points
There are 7 lines in the plane. A point is called a good point if it is contained on at least three of these seven lines. What is the maximum number of good points?