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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1963 German National Olympiad
1963 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
3
1
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sum a^2/( b^2 + c^2 ) >=3/2
It has to be proven: If at least two of the real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are different from zero, then the inequality holds
a
2
b
2
+
c
2
+
b
2
c
2
+
a
2
+
c
2
a
2
+
b
2
≥
3
2
\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32
b
2
+
c
2
a
2
+
c
2
+
a
2
b
2
+
a
2
+
b
2
c
2
≥
2
3
Under what conditions does equality occur?
2
1
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tan 2x / tan x - 2 cot 2x / cot x=1
For which numbers
x
x
x
of the interval
0
<
x
<
π
0 < x <\pi
0
<
x
<
π
holds:
tan
2
x
tan
x
−
2
cot
2
x
cot
x
=
1
\frac{\tan 2x}{\tan x} -\frac{2 \cot 2x}{\cot x}=1
tan
x
tan
2
x
−
cot
x
2
cot
2
x
=
1
1
1
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dividing a prime number by 30 or 60
a) Prove that when you divide any prime number by
30
30
30
, the remainder is either
1
1
1
or is a prime number! b) Does this also apply when dividing a prime number by
60
60
60
? Justify your answer!
6
1
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6 points lie on a sphere, starting with a tetrahedron
Consider a pyramid
A
B
C
D
ABCD
A
BC
D
whose base
A
B
C
ABC
A
BC
is a triangle. Through a point
M
M
M
of the edge
D
A
DA
D
A
, the lines
M
N
MN
MN
and
M
P
MP
MP
on the plane of the surfaces
D
A
B
DAB
D
A
B
and
D
A
C
DAC
D
A
C
are drawn respectively, such that
N
N
N
is on
D
B
DB
D
B
and
P
P
P
is on
D
C
DC
D
C
and
A
B
N
M
ABNM
A
BNM
,
A
C
P
M
ACPM
A
CPM
are cyclic quadrilaterals. a) Prove that
B
C
P
N
BCPN
BCPN
is also a cyclic quadrilateral. b) Prove that the points
A
,
B
,
C
,
M
,
N
,
P
A,B,C,M,N, P
A
,
B
,
C
,
M
,
N
,
P
lie on a sphere.
5
1
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locus of midpoints of segments of fixed length, moving on sides of square
Given is a square with side length
a
a
a
. A distance
P
Q
PQ
PQ
of length
p
p
p
, where
p
<
a
p < a
p
<
a
, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments
P
Q
PQ
PQ
?
4
1
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octagon with equal sides by removing 4 right triangles from a rectangle
Consider a rectangle with sides
2
a
2a
2
a
and
2
b
2b
2
b
, where
a
>
b
a > b
a
>
b
. There should be four congruent right triangles (one triangle at each vertex of this rectangle , whose legs are on the sides of the rectangle lie) must be cut off so that the remaining figure forms an octagon with sides of equal length. The side of the octagon is to be expressed in terms of a and
b
b
b
and constructed from
a
a
a
and
b
b
b
. Besides that it must be stated under which conditions the problem can be solved.