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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2001 Austria Beginners' Competition
2001 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
3
1
Hide problems
(x-1)^2(x-4)^2<(x-2)^2
Find all real numbers
x
x
x
such that
(
x
−
1
)
2
(
x
−
4
)
2
<
(
x
−
2
)
2
(x-1)^2(x-4)^2<(x-2)^2
(
x
−
1
)
2
(
x
−
4
)
2
<
(
x
−
2
)
2
.
2
1
Hide problems
x^2-2mx-1=0, x_1^3+x_2^3=8(x_1+x_2)
Consider the quadratic equation
x
2
−
2
m
x
−
1
=
0
x^2-2mx-1=0
x
2
−
2
m
x
−
1
=
0
, where
m
m
m
is an arbitrary real number. For what values of
m
m
m
does the equation have two real solutions, such that the sum of their cubes is equal to eight times their sum.
4
1
Hide problems
CQRP # if ABR, CBP, ACQ right isoscleles,
Let
A
B
C
ABC
A
BC
be a triangle whose angles
α
=
∠
C
A
B
\alpha=\angle CAB
α
=
∠
C
A
B
and
β
=
∠
C
B
A
\beta=\angle CBA
β
=
∠
CB
A
are greater than
4
5
∘
45^{\circ}
4
5
∘
. Above the side
A
B
AB
A
B
a right isosceles triangle
A
B
R
ABR
A
BR
is constructed with
A
B
AB
A
B
as the hypotenuse, such that
R
R
R
is inside the triangle
A
B
C
ABC
A
BC
. Analogously we construct above the sides
B
C
BC
BC
and
A
C
AC
A
C
the right isosceles triangles
C
B
P
CBP
CBP
and
A
C
Q
ACQ
A
CQ
, right at
P
P
P
and in
Q
Q
Q
, but with these outside the triangle
A
B
C
ABC
A
BC
. Prove that
C
Q
R
P
CQRP
CQRP
is a parallelogram.
1
1
Hide problems
n^n-n is divisible by 24 for odd n
Prove that for every odd positive integer
n
n
n
the number
n
n
−
n
n^n-n
n
n
−
n
is divisible by
24
24
24
.