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National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2004 Austria Beginners' Competition
2004 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
3
1
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(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0 . sum X_i^3=0
Determine the value of the parameter
m
m
m
such that the equation
(
m
−
2
)
x
2
+
(
m
2
−
4
m
+
3
)
x
−
(
6
m
2
−
2
)
=
0
(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0
(
m
−
2
)
x
2
+
(
m
2
−
4
m
+
3
)
x
−
(
6
m
2
−
2
)
=
0
has real solutions, and the sum of the third powers of these solutions is equal to zero.
2
1
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x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)} integers
For what pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
does the inequality
x
2
+
5
y
2
−
6
≤
(
x
2
−
2
)
(
y
2
−
0.04
)
x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}
x
2
+
5
y
2
−
6
≤
(
x
2
−
2
)
(
y
2
−
0.04
)
hold?
1
1
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min 4-digit , when divided by 3 gives a 4-digit number with the same digits
Find the smallest four-digit number that when divided by
3
3
3
gives a four-digit number with the same digits. (Note: Four digits means that the thousand Unit digit must not be
0
0
0
.)
4
1
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rhombus ABCD construction given circumradii of ABC, BCD
Of a rhombus
A
B
C
D
ABCD
A
BC
D
we know the circumradius
R
R
R
of
Δ
A
B
C
\Delta ABC
Δ
A
BC
and
r
r
r
of
Δ
B
C
D
\Delta BCD
Δ
BC
D
. Construct the rhombus.