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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1996 German National Olympiad
1996 German National Olympiad
Part of
German National Olympiad
Subcontests
(7)
6b
1
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each point of a plane is colored in one of three colors: red, black and blue
Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.
6a
1
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p(x) = x^3 + Ax^2 + Bx +C has 3 roots, at least 2 distinct then A^2+B^2+18C>0
Prove the following statement: If a polynomial
p
(
x
)
=
x
3
+
A
x
2
+
B
x
+
C
p(x) = x^3 + Ax^2 + Bx +C
p
(
x
)
=
x
3
+
A
x
2
+
B
x
+
C
has three real positve roots at least two of which are distinct, then
A
2
+
B
2
+
18
C
>
0
A^2 +B^2 +18C > 0
A
2
+
B
2
+
18
C
>
0
.
5
1
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points construction, intersecting chords related
Given two non-intersecting chords
A
B
AB
A
B
and
C
D
CD
C
D
of a circle
k
k
k
and a length
a
<
C
D
a <CD
a
<
C
D
. Determine a point
X
X
X
on
k
k
k
with the following property: If lines
X
A
XA
X
A
and
X
B
XB
XB
intersect
C
D
CD
C
D
at points
P
P
P
and
Q
Q
Q
respectively, then
P
Q
=
a
PQ = a
PQ
=
a
. Show how to construct all such points
X
X
X
and prove that the obtained points indeed have the desired property.
4
1
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\sqrt[3]{x^2 }+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7, x-y = 7
Find all pairs of real numbers
(
x
,
y
)
(x,y)
(
x
,
y
)
which satisfy the system
{
x
−
y
=
7
x
2
3
+
x
y
3
+
y
2
3
=
7
\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}
{
x
−
y
=
7
3
x
2
+
3
x
y
+
3
y
2
=
7
3
1
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min volume in tetrahedron
Let be given an arbitrary tetrahedron
A
B
C
D
ABCD
A
BC
D
with volume
V
V
V
. Consider all lines which pass through the barycenter
S
S
S
of the tetrahedron and intersect the edges
A
D
,
B
D
,
C
D
AD,BD,CD
A
D
,
B
D
,
C
D
at points
A
′
,
B
′
,
C
A',B',C
A
′
,
B
′
,
C
respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of
V
V
V
2
1
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a,b,>1: x < 1, y < 1, ax+by < 1, 1/(1-ax-by) \le a/(1-x)+b/(1-y) <=> a+b = 1
Let
a
a
a
and
b
b
b
be positive real numbers smaller than
1
1
1
. Prove that the following two statements are equivalent: (i)
a
+
b
=
1
a+b = 1
a
+
b
=
1
, (ii) Whenever
x
,
y
x,y
x
,
y
are positive real numbers such that
x
<
1
,
y
<
1
,
a
x
+
b
y
<
1
x < 1, y < 1, ax+by < 1
x
<
1
,
y
<
1
,
a
x
+
b
y
<
1
, the following inequlity holds:
1
1
−
a
x
−
b
y
≤
a
1
−
x
+
b
1
−
y
\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}
1
−
a
x
−
b
y
1
≤
1
−
x
a
+
1
−
y
b
1
1
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adding a few digitst o decimal writing of n can obtain 1996n
Find all natural numbers
n
n
n
with the following property: Given the decimal writing of
n
n
n
, adding a few digits one can obtain the decimal writing of
1996
n
1996n
1996
n
.