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National and Regional Contests
Greece Contests
Greece Team Selection Test
2020 Greece Team Selection Test
2020 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(2)
2
1
Hide problems
concurrency wanted, 3 circles related, a perpendiculare bisector
Given a triangle
A
B
C
ABC
A
BC
inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
(with center
O
O
O
and radius
R
R
R
) with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
and let
B
D
BD
B
D
be a diameter of the circle
c
c
c
. The perpendicular bisector of
B
D
BD
B
D
intersects line
A
C
AC
A
C
at point
M
M
M
and line
A
B
AB
A
B
at point
N
N
N
. Line
N
D
ND
N
D
intersects the circle
c
c
c
at point
T
T
T
. Let
S
S
S
be the second intersection point of cicumcircles
c
1
c_1
c
1
of triangle
O
C
M
OCM
OCM
, and
c
2
c_2
c
2
of triangle
O
A
D
OAD
O
A
D
. Prove that lines
A
D
,
C
T
AD, CT
A
D
,
CT
and
O
S
OS
OS
pass through the same point.
1
1
Hide problems
f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx , in R_+
Let
R
+
=
(
0
,
+
∞
)
R_+=(0,+\infty)
R
+
=
(
0
,
+
∞
)
. Find all functions
f
:
R
+
→
R
+
f: R_+ \to R_+
f
:
R
+
→
R
+
such that
f
(
x
f
(
y
)
)
+
f
(
y
f
(
z
)
)
+
f
(
z
f
(
x
)
)
=
x
y
+
y
z
+
z
x
f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx
f
(
x
f
(
y
))
+
f
(
y
f
(
z
))
+
f
(
z
f
(
x
))
=
x
y
+
yz
+
z
x
, for all
x
,
y
,
z
∈
R
+
x,y,z \in R_+
x
,
y
,
z
∈
R
+
.by Athanasios Kontogeorgis (aka socrates)