MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2010 Regional Competition For Advanced Students
2010 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Arithmetic progression of second order
Let
(
b
n
)
n
≥
0
=
∑
k
=
0
n
(
a
0
+
k
d
)
(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)
(
b
n
)
n
≥
0
=
∑
k
=
0
n
(
a
0
+
k
d
)
for positive integers
a
0
a_0
a
0
and
d
d
d
. We consider all such sequences containing an element
b
i
b_i
b
i
which equals
2010
2010
2010
. Determine the greatest possible value of
i
i
i
and for this value the integers
a
0
a_0
a
0
and
d
d
d
.(41th Austrian Mathematical Olympiad, regional competition, problem 4)
3
1
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∆ABC and ∆AUV are similar
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle and let
D
D
D
be a point on side
B
C
‾
\overline{BC}
BC
. Let
U
U
U
and
V
V
V
be the circumcenters of triangles
△
A
B
D
\triangle ABD
△
A
B
D
and
△
A
D
C
\triangle ADC
△
A
D
C
, respectively. Show, that
△
A
B
C
\triangle ABC
△
A
BC
and
△
A
U
V
\triangle AUV
△
A
U
V
are similar.(41th Austrian Mathematical Olympiad, regional competition, problem 3)
2
1
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Ugly equation in x,y,z
Solve the following in equation in
R
3
\mathbb{R}^3
R
3
:
4
x
4
−
x
2
(
4
y
4
+
4
z
4
−
1
)
−
2
x
y
z
+
y
8
+
2
y
4
z
4
+
y
2
z
2
+
z
8
=
0.
4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.
4
x
4
−
x
2
(
4
y
4
+
4
z
4
−
1
)
−
2
x
yz
+
y
8
+
2
y
4
z
4
+
y
2
z
2
+
z
8
=
0.
1
1
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Inequality with 0≤a,b≤1
Let
0
≤
a
0 \le a
0
≤
a
,
b
≤
1
b \le 1
b
≤
1
be real numbers. Prove the following inequality:
a
3
b
3
+
(
1
−
a
2
)
(
1
−
a
b
)
(
1
−
b
2
)
≤
1.
\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.
a
3
b
3
+
(
1
−
a
2
)
(
1
−
ab
)
(
1
−
b
2
)
≤
1.
(41th Austrian Mathematical Olympiad, regional competition, problem 1)