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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2011 German National Olympiad
2011 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
4
1
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Set of points satisfying Pythagorean-style condition
There are two points
A
A
A
and
B
B
B
in the plane. a) Determine the set
M
M
M
of all points
C
C
C
in the plane for which
∣
A
C
∣
2
+
∣
B
C
∣
2
=
2
⋅
∣
A
B
∣
2
.
|AC|^2 +|BC|^2 = 2\cdot|AB|^2.
∣
A
C
∣
2
+
∣
BC
∣
2
=
2
⋅
∣
A
B
∣
2
.
b) Decide whether there is a point
C
∈
M
C\in M
C
∈
M
such that
∠
A
C
B
\angle ACB
∠
A
CB
is maximal and if so, determine this angle.
2
1
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Can you cheat the post office?
The price for sending a packet (a rectangular cuboid) is directly proportional to the sum of its length, width, and height. Is it possible to reduce the cost of sending a packet by putting it into a cheaper packet?
6
1
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Sequence of polynomials divisible by p
Let
p
>
2
p>2
p
>
2
be a prime. Define a sequence
(
Q
n
(
x
)
)
(Q_{n}(x))
(
Q
n
(
x
))
of polynomials such that
Q
0
(
x
)
=
1
,
Q
1
(
x
)
=
x
Q_{0}(x)=1, Q_{1}(x)=x
Q
0
(
x
)
=
1
,
Q
1
(
x
)
=
x
and
Q
n
+
1
(
x
)
=
x
Q
n
(
x
)
+
n
Q
n
−
1
(
x
)
Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)
Q
n
+
1
(
x
)
=
x
Q
n
(
x
)
+
n
Q
n
−
1
(
x
)
for
n
≥
1.
n\geq 1.
n
≥
1.
Prove that
Q
p
(
x
)
−
x
p
Q_{p}(x)-x^p
Q
p
(
x
)
−
x
p
is divisible by
p
p
p
for all integers
x
.
x.
x
.
1
1
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Inequality involving Sine values
Prove for each non-negative integer
n
n
n
and real number
x
x
x
the inequality
sin
x
⋅
(
n
sin
x
−
sin
n
x
)
≥
0
\sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0
sin
x
⋅
(
n
sin
x
−
sin
n
x
)
≥
0
3
1
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Three circles for equal lenghts
Let
A
B
C
ABC
A
BC
be an acute triangle and
D
D
D
the foot of the altitude from
A
A
A
onto
B
C
BC
BC
. A semicircle with diameter
B
C
BC
BC
intersects segments
A
B
,
A
C
AB,AC
A
B
,
A
C
and
A
D
AD
A
D
in the points
F
,
E
F,E
F
,
E
resp.
X
X
X
. The circumcircles of the triangles
D
E
X
DEX
D
EX
and
D
X
F
DXF
D
XF
intersect
B
C
BC
BC
in
L
L
L
resp.
N
N
N
other than
D
D
D
. Prove
B
N
=
L
C
BN=LC
BN
=
L
C
.
5
1
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Exists prime or not?
Prove or disprove:
∃
n
∈
N
\exists n\in N
∃
n
∈
N
, s.t.
324
+
45
5
n
324 + 455^n
324
+
45
5
n
is prime.