MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2023 German National Olympiad
2023 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
6
1
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Ceiling of power of root of cubic equation is divisible by 3
The equation
x
3
−
3
x
2
+
1
=
0
x^3-3x^2+1=0
x
3
−
3
x
2
+
1
=
0
has three real solutions
x
1
<
x
2
<
x
3
x_1<x_2<x_3
x
1
<
x
2
<
x
3
. Show that for any positive integer
n
n
n
, the number
⌈
x
3
n
⌉
\left\lceil x_3^n\right\rceil
⌈
x
3
n
⌉
is a multiple of
3
3
3
.
5
1
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Symmedians of ABC and ABH intersect on line between feet of altitudes
Let
A
B
C
ABC
A
BC
be an acute triangle with altitudes
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
and orthocenter
H
H
H
. Let
C
0
C_0
C
0
be the midpoint of the segment
A
B
AB
A
B
. Let
g
g
g
be the line symmetric to the line
C
C
0
CC_0
C
C
0
with respect to the angular bisector of
∠
A
C
B
\angle ACB
∠
A
CB
. Let
h
h
h
be the line symmetric to the line
H
C
0
HC_0
H
C
0
with respect to the angular bisector of
∠
A
H
B
\angle AHB
∠
A
H
B
.Show that the lines
g
g
g
and
h
h
h
intersect on the line
A
′
B
′
A'B'
A
′
B
′
.
4
1
Hide problems
Cyclic system of two equations in three variables
Determine all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of real numbers with
a
+
4
b
=
b
+
4
c
=
c
+
4
a
.
a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.
a
+
b
4
=
b
+
c
4
=
c
+
a
4
.
3
1
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Hard combinatorics with team nominated based on three rankings
For a competition a school wants to nominate a team of
k
k
k
students, where
k
k
k
is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the
n
≥
2
n \ge 2
n
≥
2
students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition:
(
∗
)
(*)
(
∗
)
If a student
X
X
X
is not nominated for the team, there is a student
Y
Y
Y
on the team who defeated
X
X
X
in at least two disciplines.Determine all positive integers
n
≥
2
n \ge 2
n
≥
2
such that for any combination of rankings, a team can be chosen to satisfy the condition
(
∗
)
(*)
(
∗
)
, whena)
k
=
2
k=2
k
=
2
, b)
k
=
3
k=3
k
=
3
.
2
1
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Extending edges of triangle increases area by a factor of 13
In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex. Show that the area of the resulting hexagon is at least
13
13
13
times the area of the original triangle.
1
1
Hide problems
Diophantine equation in two variables with 2023
Determine all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of integers with
n
≥
m
n \ge m
n
≥
m
satisfying the equation
n
3
+
m
3
−
n
m
(
n
+
m
)
=
2023.
n^3+m^3-nm(n+m)=2023.
n
3
+
m
3
−
nm
(
n
+
m
)
=
2023.