MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2007 German National Olympiad
2007 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
5
1
Hide problems
Any two elements form an arithmetic progression
Determine all finite sets
M
M
M
of real numbers such that
M
M
M
contains at least
2
2
2
numbers and any two elements of
M
M
M
belong to an arithmetic progression of elements of
M
M
M
with three terms.
3
1
Hide problems
Similar triangles with same orientation
We say that two triangles are oriented similarly if they are similar and have the same orientation. Prove that if
A
L
T
,
A
R
M
,
O
R
T
,
ALT, ARM, ORT,
A
L
T
,
A
RM
,
ORT
,
and
U
L
M
ULM
UL
M
are four triangles which are oriented similarly, then
A
A
A
is the midpoint of the line segment
O
U
.
OU.
O
U
.
2
1
Hide problems
Product of elements in two sets is a square
Let
A
A
A
be the set of odd integers
≤
2
n
−
1.
\leq 2n-1.
≤
2
n
−
1.
For a positive integer
m
m
m
, let
B
=
{
a
+
m
∣
a
∈
A
}
.
B=\{a+m\,|\, a\in A \}.
B
=
{
a
+
m
∣
a
∈
A
}
.
Determine for which positive integers
n
n
n
there exists a positive integer
m
m
m
such that the product of all elements in
A
A
A
and
B
B
B
is a square.
1
1
Hide problems
Inequality with n-th power
Determine all real numbers
x
x
x
such that for all positive integers
n
n
n
the inequality
(
1
+
x
)
n
≤
1
+
(
2
n
−
1
)
x
(1+x)^n \leq 1+(2^n -1)x
(
1
+
x
)
n
≤
1
+
(
2
n
−
1
)
x
is true.
4
1
Hide problems
Unusual triangle as potential IMO logo.
Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.
6
1
Hide problems
Inequality from equality
For two real numbers a,b the equation:
x
4
−
a
x
3
+
6
x
2
−
b
x
+
1
=
0
x^{4}-ax^{3}+6x^{2}-bx+1=0
x
4
−
a
x
3
+
6
x
2
−
b
x
+
1
=
0
has four solutions (not necessarily distinct). Prove that
a
2
+
b
2
≥
32
a^{2}+b^{2}\ge{32}
a
2
+
b
2
≥
32