MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2006 Federal Competition For Advanced Students, Part 2
2006 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
Hide problems
37th Austrian Mathematical Olympiad 2006
The triangle
A
B
C
ABC
A
BC
is given. On the extension of the side
A
B
AB
A
B
we construct the point
R
R
R
with BR \equal{} BC, where
A
R
>
B
R
AR > BR
A
R
>
BR
and on the extension of the side
A
C
AC
A
C
we construct the point
S
S
S
with CS \equal{} CB, where
A
S
>
C
S
AS > CS
A
S
>
CS
. Let
A
1
A_1
A
1
be the point of intersection of the diagonals of the quadrilateral
B
R
S
C
BRSC
BRSC
. Analogous we construct the point
T
T
T
on the extension of the side
B
C
BC
BC
, where CT \equal{} CA and
B
T
>
C
T
BT > CT
BT
>
CT
and on the extension of the side
B
A
BA
B
A
we construct the point
U
U
U
with AU \equal{} AC, where
B
U
>
A
U
BU > AU
B
U
>
A
U
. Let
B
1
B_1
B
1
be the point of intersection of the diagonals of the quadrilateral
C
T
U
A
CTUA
CT
U
A
. Likewise we construct the point
V
V
V
on the extension of the side
C
A
CA
C
A
, where AV \equal{} AB and
C
V
>
A
V
CV > AV
C
V
>
A
V
and on the extension of the side
C
B
CB
CB
we construct the point
W
W
W
with BW \equal{} BA and
C
W
>
B
W
CW > BW
C
W
>
B
W
. Let
C
1
C_1
C
1
be the point of intersection of the diagonals of the quadrilateral
A
V
W
B
AVWB
A
VW
B
. Show that the area of the hexagon
A
C
1
B
A
1
C
B
1
AC_1BA_1CB_1
A
C
1
B
A
1
C
B
1
is equal to the sum of the areas of the triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
.
37th Austrian Mathematical Olympiad 2006
Let
A
A
A
be an integer not equal to
0
0
0
. Solve the following system of equations in
Z
3
\mathbb{Z}^3
Z
3
. x \plus{} y^2 \plus{} z^3 \equal{} A \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A} xy^2z^3 \equal{} A^2
2
2
Hide problems
37th Austrian Mathematical Olympiad 2006
Find all monotonous functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
that satisfy the following functional equation: f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.
37th Austrian Mathematical Olympiad 2006
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Show that 3(a \plus{} b \plus{} c) \ge 8 \sqrt [3]{abc} \plus{} \sqrt [3]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3} }.
1
2
Hide problems
37th Austrian Mathematical Olympiad 2006
Let
N
N
N
be a positive integer. How many non-negative integers
n
≤
N
n \le N
n
≤
N
are there that have an integer multiple, that only uses the digits
2
2
2
and
6
6
6
in decimal representation?
37th Austrian Mathematical Olympiad 2006
For which rational
x
x
x
is the number 1 \plus{} 105 \cdot 2^x the square of a rational number?