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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2011 Greece National Olympiad
2011 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
2
1
Hide problems
Good Points and Segments
In the Cartesian plane
O
x
y
Oxy
O
x
y
we consider the points
A
1
(
40
,
1
)
,
A
2
(
40
,
2
)
,
…
,
A
40
(
40
,
40
)
{A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)
A
1
(
40
,
1
)
,
A
2
(
40
,
2
)
,
…
,
A
40
(
40
,
40
)
as well as the segments
O
A
1
,
O
A
2
,
…
,
O
A
40
O{A_1},O{A_2},\ldots,O{A_{40}}
O
A
1
,
O
A
2
,
…
,
O
A
40
. A point of the Cartesian plane
O
x
y
Oxy
O
x
y
is called "good", if its coordinates are integers and it is internal of one segment
O
A
i
,
i
=
1
,
2
,
3
,
…
,
40
O{A_i}, i=1,2,3,\ldots,40
O
A
i
,
i
=
1
,
2
,
3
,
…
,
40
. Additionally, one of the segments
O
A
1
,
O
A
2
,
…
,
O
A
40
O{A_1},O{A_2},\ldots,O{A_{40}}
O
A
1
,
O
A
2
,
…
,
O
A
40
is called "good" if it contains a "good" point. Find the number of "good" segments and "good" points.
1
1
Hide problems
Solve the equation x^3y^2(2y - x) = x^2y^4-36
Solve in integers the equation
x
3
y
2
(
2
y
−
x
)
=
x
2
y
4
−
36
{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36
x
3
y
2
(
2
y
−
x
)
=
x
2
y
4
−
36
3
1
Hide problems
Inequality from Greece
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with sum
6
6
6
. Find the maximum value of
S
=
a
2
+
2
b
c
3
+
b
2
+
2
c
a
3
+
c
2
+
2
a
b
3
.
S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.
S
=
3
a
2
+
2
b
c
+
3
b
2
+
2
c
a
+
3
c
2
+
2
ab
.
4
1
Hide problems
MZ is perp to BC iff CA=CB or Z ≡ O
We consider an acute angled triangle
A
B
C
ABC
A
BC
(with
A
B
<
A
C
AB<AC
A
B
<
A
C
) and its circumcircle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
(with center
O
O
O
and semidiametre
R
R
R
).The altitude
A
D
AD
A
D
cuts the circumcircle at the point
E
E
E
,while the perpedicular bisector
(
m
)
(m)
(
m
)
of the segment
A
B
AB
A
B
,cuts
A
D
AD
A
D
at the point
L
L
L
.
B
L
BL
B
L
cuts
A
C
AC
A
C
at the point
M
M
M
and the circumcircle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at the point
N
N
N
.Finally
E
N
EN
EN
cuts the perpedicular bisector
(
m
)
(m)
(
m
)
at the point
Z
Z
Z
.Prove that:
M
Z
⊥
B
C
⟺
(
C
A
=
C
B
or
Z
≡
O
)
MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right)
MZ
⊥
BC
⟺
(
C
A
=
CB
or
Z
≡
O
)