MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2000 German National Olympiad
2000 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
5
1
Hide problems
2n distinct points on a circumference, n red and n blue
(a) Let be given
2
n
2n
2
n
distinct points on a circumference,
n
n
n
of which are red and
n
n
n
are blue. Prove that one can join these points pairwise by
n
n
n
segments so that no two segments intersect and the endpoints of each segments have different colors. (b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.
6
1
Hide problems
a_{2^m} = 1/m, a_{2n-1}a_{2n} = a_n, a_{2n}a_{2n+1} = a_{2^m+n}
A sequence (
a
n
a_n
a
n
) satisfies the following conditions: (i) For each
m
∈
N
m \in N
m
∈
N
it holds that
a
2
m
=
1
/
m
a_{2^m} = 1/m
a
2
m
=
1/
m
. (ii) For each natural
n
≥
2
n \ge 2
n
≥
2
it holds that
a
2
n
−
1
a
2
n
=
a
n
a_{2n-1}a_{2n} = a_n
a
2
n
−
1
a
2
n
=
a
n
. (iii) For all integers
m
,
n
m,n
m
,
n
with
2
m
>
n
≥
1
2m > n \ge 1
2
m
>
n
≥
1
it holds that
a
2
n
a
2
n
+
1
=
a
2
m
+
n
a_{2n}a_{2n+1} = a_{2^m+n}
a
2
n
a
2
n
+
1
=
a
2
m
+
n
. Determine
a
2000
a_{2000}
a
2000
. You may assume that such a sequence exists.
4
1
Hide problems
\sqrt{x+y}+\sqrt{z} = 7, \sqrt{x+z}+\sqrt{y} = 7,\sqrt{y+z}+\sqrt{x} = 5
Find all nonnegative solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
to the system
{
x
+
y
+
z
=
7
x
+
z
+
y
=
7
y
+
z
+
x
=
5
\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\ \sqrt{x+z}+\sqrt{y} = 7 \\ \sqrt{y+z}+\sqrt{x} = 5 \end{cases}
⎩
⎨
⎧
x
+
y
+
z
=
7
x
+
z
+
y
=
7
y
+
z
+
x
=
5
3
1
Hide problems
<BAO, < CBO, < ACO >= 30^o, O interior of ABC => ABC equilateral
Suppose that an interior point
O
O
O
of a triangle
A
B
C
ABC
A
BC
is such that the angles
∠
B
A
O
,
∠
C
B
O
,
∠
A
C
O
\angle BAO,\angle CBO, \angle ACO
∠
B
A
O
,
∠
CBO
,
∠
A
CO
are all greater than or equal to
3
0
o
30^o
3
0
o
. Prove that the triangle
A
B
C
ABC
A
BC
is equilateral.
2
1
Hide problems
min of f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4
For an integer
n
≥
2
n \ge 2
n
≥
2
, find all real numbers
x
x
x
for which the polynomial
f
(
x
)
=
(
x
−
1
)
4
+
(
x
−
2
)
4
+
.
.
.
+
(
x
−
n
)
4
f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4
f
(
x
)
=
(
x
−
1
)
4
+
(
x
−
2
)
4
+
...
+
(
x
−
n
)
4
takes its minimum value.
1
1
Hide problems
|x|+|y| = 1, x^2 +y^2 = a
For each real parameter
a
a
a
, find the number of real solutions to the system
{
∣
x
∣
+
∣
y
∣
=
1
,
x
2
+
y
2
=
a
\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}
{
∣
x
∣
+
∣
y
∣
=
1
,
x
2
+
y
2
=
a