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Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2009 Greece Team Selection Test
2009 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
4
1
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N coloured points on the plane...
Given are
N
N
N
points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a "value" according to the following conditions: i.If at least one of the endpoints of a segment is black then the segment's "value" is
0
0
0
. ii.If the endpoints of the segment have the same colour,re or green,then the segment's "value" is
1
1
1
. iii.If the endpoints of the segment have different colours but none of them is black,then the segment's "value" is
−
1
-1
−
1
.Determine the minimum possible sum of the "values" of the segments.
3
1
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Find all triples of reals...
Find all triples
(
x
,
y
,
z
)
∈
R
3
(x,y,z)\in \mathbb{R}^{3}
(
x
,
y
,
z
)
∈
R
3
such that
x
,
y
,
z
>
3
x,y,z>3
x
,
y
,
z
>
3
and
(
x
+
2
)
2
y
+
z
−
2
+
(
y
+
4
)
2
z
+
x
−
4
+
(
z
+
6
)
2
x
+
y
−
6
=
36
\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36
y
+
z
−
2
(
x
+
2
)
2
+
z
+
x
−
4
(
y
+
4
)
2
+
x
+
y
−
6
(
z
+
6
)
2
=
36
2
1
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My circumcenter is your barycenter
Given is a triangle
A
B
C
ABC
A
BC
with barycenter
G
G
G
and circumcenter
O
O
O
.The perpendicular bisectors of
G
A
,
G
B
,
G
C
GA,GB,GC
G
A
,
GB
,
GC
intersect at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
.Show that
O
O
O
is the barycenter of
△
A
1
B
1
C
1
\triangle{A_1B_1C_1}
△
A
1
B
1
C
1
.
1
1
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Perfect square gives divisibility
Suppose that
a
a
a
is an even positive integer and
A
=
a
n
+
a
n
−
1
+
…
+
a
+
1
,
n
∈
N
∗
A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}
A
=
a
n
+
a
n
−
1
+
…
+
a
+
1
,
n
∈
N
∗
is a perfect square.Prove that
8
∣
a
8\mid a
8
∣
a
.