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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2009 Greece National Olympiad
2009 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Geometry with complex numbers
Consider pairwise distinct complex numbers
z
1
,
z
2
,
z
3
,
z
4
,
z
5
,
z
6
z_1,z_2,z_3,z_4,z_5,z_6
z
1
,
z
2
,
z
3
,
z
4
,
z
5
,
z
6
whose images
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
A_1,A_2,A_3,A_4,A_5,A_6
A
1
,
A
2
,
A
3
,
A
4
,
A
5
,
A
6
respectively are succesive points on the circle centered at
O
(
0
,
0
)
O(0,0)
O
(
0
,
0
)
and having radius
r
>
0.
r>0.
r
>
0.
If
w
w
w
is a root of the equation
z
2
+
z
+
1
=
0
z^2+z+1=0
z
2
+
z
+
1
=
0
and the next equalities hold
z
1
w
2
+
z
3
w
+
z
5
=
0
z
2
w
2
+
z
4
w
+
z
6
=
0
z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0
z
1
w
2
+
z
3
w
+
z
5
=
0
z
2
w
2
+
z
4
w
+
z
6
=
0
prove thata) Triangle
A
1
A
3
A
5
A_1A_3A_5
A
1
A
3
A
5
is equilateral b)
∣
z
1
−
z
2
∣
+
∣
z
2
−
z
3
∣
+
∣
z
3
−
z
4
∣
+
∣
z
4
−
z
5
∣
+
z
5
−
z
6
∣
+
∣
z
6
−
z
1
∣
=
3
∣
z
1
−
z
4
∣
=
3
∣
z
2
−
z
5
∣
=
3
∣
z
3
−
z
6
∣
.
|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.
∣
z
1
−
z
2
∣
+
∣
z
2
−
z
3
∣
+
∣
z
3
−
z
4
∣
+
∣
z
4
−
z
5
∣
+
z
5
−
z
6
∣
+
∣
z
6
−
z
1
∣
=
3∣
z
1
−
z
4
∣
=
3∣
z
2
−
z
5
∣
=
3∣
z
3
−
z
6
∣.
2
1
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Concurrent lines
Consider a triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
and let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the midpoints of the sides
B
C
,
A
C
,
A
B
,
BC,AC,AB,
BC
,
A
C
,
A
B
,
respectively. Points
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
are defined as
O
A
2
→
=
λ
⋅
O
A
1
→
,
O
B
2
→
=
λ
⋅
O
B
1
→
,
O
C
2
→
=
λ
⋅
O
C
1
→
,
\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},
O
A
2
=
λ
⋅
O
A
1
,
O
B
2
=
λ
⋅
O
B
1
,
O
C
2
=
λ
⋅
O
C
1
,
where
λ
>
0.
\lambda >0.
λ
>
0.
Prove that lines
A
A
2
,
B
B
2
,
C
C
2
AA_2,BB_2,CC_2
A
A
2
,
B
B
2
,
C
C
2
are concurrent.
1
1
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Integer and rational
Find all positive integers
n
n
n
such that the number
A
=
9
n
−
1
n
+
7
A=\sqrt{\frac{9n-1}{n+7}}
A
=
n
+
7
9
n
−
1
is rational.
3
1
Hide problems
greek mathematical olympiad 2009
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be nonnegative real numbers such that x \plus{} y \plus{} z \equal{} 2. Prove that x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1. When does the equality occur?