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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1997 German National Olympiad
1997 German National Olympiad
Part of
German National Olympiad
Subcontests
(7)
6b
1
Hide problems
this procedure must end in a finite time for any choice of e and points P_i
An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon
P
1
P
2
P
3
P
4
P
5
P_1P_2P_3P_4P_5
P
1
P
2
P
3
P
4
P
5
in a circle. Now choose an arror bound
ϵ
>
0
\epsilon > 0
ϵ
>
0
and apply the following procedure. (a) Denote
P
0
=
P
5
P_0 = P_5
P
0
=
P
5
and
P
6
=
P
1
P_6 = P_1
P
6
=
P
1
and construct the midpoint
Q
i
Q_i
Q
i
of the circular arc
P
i
−
1
P
i
+
1
P_{i-1}P_{i+1}
P
i
−
1
P
i
+
1
containing
P
i
P_i
P
i
. (b) Rename the vertices
Q
1
,
.
.
.
,
Q
5
Q_1,...,Q_5
Q
1
,
...
,
Q
5
as
P
1
,
.
.
.
,
P
5
P_1,...,P_5
P
1
,
...
,
P
5
. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs
P
i
P
i
+
1
P_iP_{i+1}
P
i
P
i
+
1
is less than
ϵ
\epsilon
ϵ
. Prove this procedure must end in a finite time for any choice of
ϵ
\epsilon
ϵ
and the points
P
i
P_i
P
i
.
6a
1
Hide problems
p divides f(x)=x^5 +5x^4 +5x^3 +5x^2 +1, g(x)=x^5 +5x^4 +3x^3 -5x^2 -1
Let us define
f
f
f
and
g
g
g
by
f
(
x
)
=
x
5
+
5
x
4
+
5
x
3
+
5
x
2
+
1
f(x) = x^5 +5x^4 +5x^3 +5x^2 +1
f
(
x
)
=
x
5
+
5
x
4
+
5
x
3
+
5
x
2
+
1
,
g
(
x
)
=
x
5
+
5
x
4
+
3
x
3
−
5
x
2
−
1
g(x) = x^5 +5x^4 +3x^3 -5x^2 -1
g
(
x
)
=
x
5
+
5
x
4
+
3
x
3
−
5
x
2
−
1
. Determine all prime numbers
p
p
p
such that, for at least one integer
x
,
0
≤
x
<
p
−
1
x, 0 \le x < p-1
x
,
0
≤
x
<
p
−
1
, both
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
are divisible by
p
p
p
. For each such
p
p
p
, find all
x
x
x
with this property.
5
1
Hide problems
n discs in plane, possibly overlapping, whose union has the area 1
We are given
n
n
n
discs in a plane, possibly overlapping, whose union has the area
1
1
1
. Prove that we can choose some of them which are mutually disjoint and have the total area greater than
1
/
9
1/9
1/9
.
4
1
Hide problems
x^3 = 2y-1, y^3 = 2z-1, z^3 = 2x-1
Find all real solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the system of equations
{
x
3
=
2
y
−
1
y
3
=
2
z
−
1
z
3
=
2
x
−
1
\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}
⎩
⎨
⎧
x
3
=
2
y
−
1
y
3
=
2
z
−
1
z
3
=
2
x
−
1
3
1
Hide problems
angle chasing inside a quadrilateral
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
we are given that
∠
C
B
D
=
1
0
o
\angle CBD = 10^o
∠
CB
D
=
1
0
o
,
∠
C
A
D
=
2
0
o
\angle CAD = 20^o
∠
C
A
D
=
2
0
o
,
∠
A
B
D
=
4
0
o
\angle ABD = 40^o
∠
A
B
D
=
4
0
o
,
∠
B
A
C
=
5
0
o
\angle BAC = 50^o
∠
B
A
C
=
5
0
o
. Determine the angles
∠
B
C
D
\angle BCD
∠
BC
D
and
∠
A
D
C
\angle ADC
∠
A
D
C
.
2
1
Hide problems
1/2^n \sum_{k = 1}^{2^n} \frac{u(k)}{k}> 2/3, greatest odd divisor
For a positive integer
k
k
k
, let us denote by
u
(
k
)
u(k)
u
(
k
)
the greatest odd divisor of
k
k
k
. Prove that, for each
n
∈
N
n \in N
n
∈
N
,
1
2
n
∑
k
=
1
2
n
u
(
k
)
k
>
2
3
\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}
2
n
1
∑
k
=
1
2
n
k
u
(
k
)
>
3
2
.
1
1
Hide problems
no perfect squares a,b,c such that ab-bc = a
Prove that there are no perfect squares
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
b
−
b
c
=
a
ab-bc = a
ab
−
b
c
=
a
.