MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2012 German National Olympiad
2012 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
3
1
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Tangents to circumcircles in a triangle
Let
A
B
C
ABC
A
BC
a triangle and
k
k
k
a circle such that: (1) The circle
k
k
k
passes through
A
A
A
and
B
B
B
and touches the line
A
C
.
AC.
A
C
.
(2) The tangent to
k
k
k
at
B
B
B
intersects the line
A
C
AC
A
C
in a point
X
≠
C
.
X\ne C.
X
=
C
.
(3) The circumcircle
ω
\omega
ω
of
B
X
C
BXC
BXC
intersects
k
k
k
in a point
Q
≠
B
.
Q\ne B.
Q
=
B
.
(4) The tangent to
ω
\omega
ω
at
X
X
X
intersects the line
A
B
AB
A
B
in a point
Y
.
Y.
Y
.
Prove that the line
X
Y
XY
X
Y
is tangent to the circumcircle of
B
Q
Y
.
BQY.
BQ
Y
.
4
1
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A fairly inelegant inequality
Let
a
,
b
a,b
a
,
b
be positive real numbers and
n
≥
2
n\geq 2
n
≥
2
a positive integer. Prove that if
x
n
≤
a
x
+
b
x^n \leq ax+b
x
n
≤
a
x
+
b
holds for a positive real number
x
x
x
, then it also satisfies the inequality
x
<
2
a
n
−
1
+
2
b
n
.
x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.
x
<
n
−
1
2
a
+
n
2
b
.
1
1
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Standard sequence contains infinitely many composite integers
Define a sequence
(
a
n
)
(a_n)
(
a
n
)
by
a
0
=
−
4
,
a
1
=
−
7
a_0 =-4 , a_1 =-7
a
0
=
−
4
,
a
1
=
−
7
and
a
n
+
2
=
5
a
n
+
1
−
6
a
n
a_{n+2}= 5a_{n+1} -6a_n
a
n
+
2
=
5
a
n
+
1
−
6
a
n
for
n
≥
0.
n\geq 0.
n
≥
0.
Prove that there are infinitely many positive integers
n
n
n
such that
a
n
a_n
a
n
is composite.
5
1
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inradius of tetrahedron, inequality with lengths of nonadjacent edges
Let
a
,
b
a,b
a
,
b
be the lengths of two nonadjacent edges of a tetrahedron with inradius
r
r
r
. Prove that
r
<
a
b
2
(
a
+
b
)
.
r<\frac{ab}{2(a+b)}.
r
<
2
(
a
+
b
)
ab
.
2
1
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Graph theory
Find the maximal number of edges a connected graph
G
G
G
with
n
n
n
vertices may have, so that after deleting an arbitrary cycle,
G
G
G
is not connected anymore.
6
1
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Algebra Problem
Let
a
1
a_1
a
1
and
a
2
a_2
a
2
be postive real numbers. Let
a
n
+
2
=
1
+
a
n
+
1
a
n
a_{n+2}=1+\frac{a_{n+1}}{a_{n}}
a
n
+
2
=
1
+
a
n
a
n
+
1
Prove that
∣
a
2012
−
2
∣
<
1
0
−
200
|a_{2012}-2|<10^{-200}
∣
a
2012
−
2∣
<
1
0
−
200