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Problems
Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2010 Greece Team Selection Test
2010 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
2
1
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Circles and Numbers
In a blackboard there are
K
K
K
circles in a row such that one of the numbers
1
,
.
.
.
,
K
1,...,K
1
,
...
,
K
is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor
d
d
d
of
K
K
K
,
1
≤
d
≤
K
1\leq d\leq K
1
≤
d
≤
K
we change the situation of the circles in which their assigned numbers are divisible by
d
d
d
,performing for each divisor
d
d
d
K
K
K
changes of situation. Determine the value of
K
K
K
for which the following holds;when this procedure is applied once for all positive divisors of
K
K
K
,then all numbers
1
,
2
,
3
,
.
.
.
,
K
1,2,3,...,K
1
,
2
,
3
,
...
,
K
are written in the circles they were assigned in.
3
1
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Concurrencies
Let
A
B
C
ABC
A
BC
be a triangle,
O
O
O
its circumcenter and
R
R
R
the radius of its circumcircle.Denote by
O
1
O_{1}
O
1
the symmetric of
O
O
O
with respect to
B
C
BC
BC
,
O
2
O_{2}
O
2
the symmetric of
O
O
O
with respect to
A
C
AC
A
C
and by
O
3
O_{3}
O
3
the symmetric of
O
O
O
with respect to
A
B
AB
A
B
. (a)Prove that the circles
C
1
(
O
1
,
R
)
C_{1}(O_{1},R)
C
1
(
O
1
,
R
)
,
C
2
(
O
2
,
R
)
C_{2}(O_{2},R)
C
2
(
O
2
,
R
)
,
C
3
(
O
3
,
R
)
C_{3}(O_{3},R)
C
3
(
O
3
,
R
)
have a common point. (b)Denote by
T
T
T
this point.Let
l
l
l
be an arbitary line passing through
T
T
T
which intersects
C
1
C_{1}
C
1
at
L
L
L
,
C
2
C_{2}
C
2
at
M
M
M
and
C
3
C_{3}
C
3
at
K
K
K
.From
K
,
L
,
M
K,L,M
K
,
L
,
M
drop perpendiculars to
A
B
,
B
C
,
A
C
AB,BC,AC
A
B
,
BC
,
A
C
respectively.Prove that these perpendiculars pass through a point.
1
1
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Easy system with positive reals
Solve in positive reals the system:
x
+
y
+
z
+
w
=
4
x+y+z+w=4
x
+
y
+
z
+
w
=
4
1
x
+
1
y
+
1
z
+
1
w
=
5
−
1
x
y
z
w
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}
x
1
+
y
1
+
z
1
+
w
1
=
5
−
x
yz
w
1
4
1
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Functional Equation
Find all functions
f
:
R
∗
→
R
∗
f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}
f
:
R
∗
→
R
∗
satisfying
f
(
f
(
x
)
f
(
y
)
)
=
1
y
f
(
f
(
x
)
)
f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))
f
(
f
(
y
)
f
(
x
)
)
=
y
1
f
(
f
(
x
))
for all
x
,
y
∈
R
∗
x,y\in \mathbb{R^{\ast }}
x
,
y
∈
R
∗
and are strictly monotone in
(
0
,
+
∞
)
(0,+\infty )
(
0
,
+
∞
)