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National and Regional Contests
Greece Contests
Greece Team Selection Test
2015 Greece Team Selection Test
2015 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
3
1
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Nice geometry with many circles
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
<
B
C
\displaystyle{AB<AC<BC}
A
B
<
A
C
<
BC
inscribed in circle
c
(
O
,
R
)
\displaystyle{c(O,R)}
c
(
O
,
R
)
.The excircle
(
c
A
)
\displaystyle{(c_A)}
(
c
A
)
has center
I
\displaystyle{I}
I
and touches the sides
B
C
,
A
C
,
A
B
\displaystyle{BC,AC,AB}
BC
,
A
C
,
A
B
of the triangle
A
B
C
ABC
A
BC
at
D
,
E
,
Z
\displaystyle{D,E,Z}
D
,
E
,
Z
respectively.
A
I
\displaystyle{AI}
A
I
cuts
(
c
)
\displaystyle{(c)}
(
c
)
at point
M
M
M
and the circumcircle
(
c
1
)
\displaystyle{(c_1)}
(
c
1
)
of triangle
A
Z
E
\displaystyle{AZE}
A
ZE
cuts
(
c
)
\displaystyle{(c)}
(
c
)
at
K
K
K
.The circumcircle
(
c
2
)
\displaystyle{(c_2)}
(
c
2
)
of the triangle
O
K
M
\displaystyle{OKM}
O
K
M
cuts
(
c
1
)
\displaystyle{(c_1)}
(
c
1
)
at point
N
N
N
.Prove that the point of intersection of the lines
A
N
,
K
I
AN,KI
A
N
,
K
I
lies on
(
c
)
\displaystyle{(c)}
(
c
)
.
4
1
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Functional inequality
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
which satisfy
y
f
(
x
)
+
f
(
y
)
≥
f
(
x
y
)
yf(x)+f(y) \geq f(xy)
y
f
(
x
)
+
f
(
y
)
≥
f
(
x
y
)
1
1
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Equation in positive integers
Solve in positive integers the following equation;
x
y
(
x
+
y
−
10
)
−
3
x
2
−
2
y
2
+
21
x
+
16
y
=
60
xy(x+y-10)-3x^2-2y^2+21x+16y=60
x
y
(
x
+
y
−
10
)
−
3
x
2
−
2
y
2
+
21
x
+
16
y
=
60
2
1
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Length of segments
Consider
111
111
111
distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least
1998
1998
1998
segments formed by these points with length
≤
3
\leq \sqrt{3}
≤
3