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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2005 Federal Competition For Advanced Students, Part 2
2005 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
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Altitudes and circles
Triangle
D
E
F
DEF
D
EF
is acute. Circle
c
1
c_1
c
1
is drawn with
D
F
DF
D
F
as its diameter and circle
c
2
c_2
c
2
is drawn with
D
E
DE
D
E
as its diameter. Points
Y
Y
Y
and
Z
Z
Z
are on
D
F
DF
D
F
and
D
E
DE
D
E
respectively so that
E
Y
EY
E
Y
and
F
Z
FZ
FZ
are altitudes of triangle
D
E
F
DEF
D
EF
.
E
Y
EY
E
Y
intersects
c
1
c_1
c
1
at
P
P
P
, and
F
Z
FZ
FZ
intersects
c
2
c_2
c
2
at
Q
Q
Q
.
E
Y
EY
E
Y
extended intersects
c
1
c_1
c
1
at
R
R
R
, and
F
Z
FZ
FZ
extended intersects
c
2
c_2
c
2
at
S
S
S
. Prove that
P
P
P
,
Q
Q
Q
,
R
R
R
, and
S
S
S
are concyclic points.
AQ = BQ [intersections of lines through Q with cube surface]
Let
Q
Q
Q
be a point inside a cube. Prove that there are infinitely many lines
l
l
l
so that
A
Q
=
B
Q
AQ=BQ
A
Q
=
BQ
where
A
A
A
and
B
B
B
are the two points of intersection of
l
l
l
and the surface of the cube.
2
2
Hide problems
Austrian MO 2005 inequality
Prove that for all positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, we have
a
+
b
+
c
+
d
a
b
c
d
≤
1
a
3
+
1
b
3
+
1
c
3
+
1
d
3
\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}
ab
c
d
a
+
b
+
c
+
d
≤
a
3
1
+
b
3
1
+
c
3
1
+
d
3
1
(a,b,c,d,e,f)
Find all real
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
that satisfy the system
4
a
=
(
b
+
c
+
d
+
e
)
4
4a = (b + c + d + e)^4
4
a
=
(
b
+
c
+
d
+
e
)
4
4
b
=
(
c
+
d
+
e
+
f
)
4
4b = (c + d + e + f)^4
4
b
=
(
c
+
d
+
e
+
f
)
4
4
c
=
(
d
+
e
+
f
+
a
)
4
4c = (d + e + f + a)^4
4
c
=
(
d
+
e
+
f
+
a
)
4
4
d
=
(
e
+
f
+
a
+
b
)
4
4d = (e + f + a + b)^4
4
d
=
(
e
+
f
+
a
+
b
)
4
4
e
=
(
f
+
a
+
b
+
c
)
4
4e = (f + a + b + c)^4
4
e
=
(
f
+
a
+
b
+
c
)
4
4
f
=
(
a
+
b
+
c
+
d
)
4
4f = (a + b + c + d)^4
4
f
=
(
a
+
b
+
c
+
d
)
4
1
2
Hide problems
LCM(a,b,c) = a+b+c
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of natural numbers, such that
L
C
M
(
a
,
b
,
c
)
=
a
+
b
+
c
LCM(a,b,c)=a+b+c
L
CM
(
a
,
b
,
c
)
=
a
+
b
+
c
f(2x+1)=f(2x), f(3x+1)=f(3x), f(5x+1)=f(5x)
The function
f
:
(
0
,
.
.
.
2005
)
→
N
f : (0,...2005) \rightarrow N
f
:
(
0
,
...2005
)
→
N
has the properties that
f
(
2
x
+
1
)
=
f
(
2
x
)
f(2x+1)=f(2x)
f
(
2
x
+
1
)
=
f
(
2
x
)
,
f
(
3
x
+
1
)
=
f
(
3
x
)
f(3x+1)=f(3x)
f
(
3
x
+
1
)
=
f
(
3
x
)
and
f
(
5
x
+
1
)
=
f
(
5
x
)
f(5x+1)=f(5x)
f
(
5
x
+
1
)
=
f
(
5
x
)
with
x
∈
(
0
,
1
,
2
,
.
.
.
,
2005
)
x \in (0,1,2,...,2005)
x
∈
(
0
,
1
,
2
,
...
,
2005
)
. How many different values can the function assume?