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National and Regional Contests
Germany Contests
Germany Team Selection Test
2002 Germany Team Selection Test
2002 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
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x^{2y} + (x+1)^{2y} = (x+2)^{2y}
Determine all
(
x
,
y
)
∈
N
2
(x,y) \in \mathbb{N}^2
(
x
,
y
)
∈
N
2
which satisfy
x
2
y
+
(
x
+
1
)
2
y
=
(
x
+
2
)
2
y
.
x^{2y} + (x+1)^{2y} = (x+2)^{2y}.
x
2
y
+
(
x
+
1
)
2
y
=
(
x
+
2
)
2
y
.
2
1
Hide problems
Lengths of the angle bisectors of a triangle with perimeter
Prove: If
x
,
y
,
z
x, y, z
x
,
y
,
z
are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
1
x
2
+
1
y
2
+
1
z
2
≥
1.
\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.
x
2
1
+
y
2
1
+
z
2
1
≥
1.
1
2
Hide problems
Divisible by 121
Determine the number of all numbers which are represented as
x
2
+
y
2
x^2+y^2
x
2
+
y
2
with
x
,
y
∈
{
1
,
2
,
3
,
…
,
1000
}
x, y \in \{1, 2, 3, \ldots, 1000\}
x
,
y
∈
{
1
,
2
,
3
,
…
,
1000
}
and which are divisible by 121.
Two-variable function
Let
P
P
P
denote the set of all ordered pairs
(
p
,
q
)
\left(p,q\right)
(
p
,
q
)
of nonnegative integers. Find all functions
f
:
P
→
R
f: P \rightarrow \mathbb{R}
f
:
P
→
R
satisfying f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.