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Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2010 Germany Team Selection Test
2010 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
2
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3^m-7^n=2
Determine all
(
m
,
n
)
∈
Z
+
×
Z
+
(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+
(
m
,
n
)
∈
Z
+
×
Z
+
which satisfy
3
m
−
7
n
=
2.
3^m-7^n=2.
3
m
−
7
n
=
2.
f(x)f(y) = (x+y+1)^2 * f((xy-1)/(x+y+1))
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
=
(
x
+
y
+
1
)
2
⋅
f
(
x
y
−
1
x
+
y
+
1
)
f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)
f
(
x
)
f
(
y
)
=
(
x
+
y
+
1
)
2
⋅
f
(
x
+
y
+
1
x
y
−
1
)
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
with
x
+
y
+
1
≠
0
x+y+1 \neq 0
x
+
y
+
1
=
0
and
f
(
x
)
>
1
f(x) > 1
f
(
x
)
>
1
∀
x
>
0.
\forall x > 0.
∀
x
>
0.
2
3
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(4a+b)/(a+4b) + (4b+c)/(b+4c) + (4c+a)/(c+4a)
Prove or disprove that
∀
a
,
b
,
c
,
d
∈
R
+
\forall a,b,c,d \in \mathbb{R}^+
∀
a
,
b
,
c
,
d
∈
R
+
we have the following inequality:
3
≤
4
a
+
b
a
+
4
b
+
4
b
+
c
b
+
4
c
+
4
c
+
a
c
+
4
a
<
33
4
3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}
3
≤
a
+
4
b
4
a
+
b
+
b
+
4
c
4
b
+
c
+
c
+
4
a
4
c
+
a
<
4
33
Number of solutions divisible by 10
We are given
m
,
n
∈
Z
+
.
m,n \in \mathbb{Z}^+.
m
,
n
∈
Z
+
.
Show the number of solution
4
−
4-
4
−
tuples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
of the system\begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*}is divisible by 10.
Regular hexagons in a non-overlapping way
Determine all
n
∈
Z
+
n \in \mathbb{Z}^+
n
∈
Z
+
such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many
n
−
n-
n
−
gons such that they can be composed into
n
n
n
congruent regular hexagons in a non-overlapping way upon certain rotations and translations.
1
5
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