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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2017 Greece National Olympiad
2017 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Coefficients of Polynomial
Let
u
u
u
be the positive root of the equation
x
2
+
x
−
4
=
0
x^2+x-4=0
x
2
+
x
−
4
=
0
. The polynomial
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
0
where
n
n
n
is positive integer has non-negative integer coefficients and
P
(
u
)
=
2017
P(u)=2017
P
(
u
)
=
2017
. 1) Prove that
a
0
+
a
1
+
.
.
.
+
a
n
≡
1
m
o
d
2
a_0+a_1+...+a_n\equiv 1\mod 2
a
0
+
a
1
+
...
+
a
n
≡
1
mod
2
. 2) Find the minimum possible value of
a
0
+
a
1
+
.
.
.
+
a
n
a_0+a_1+...+a_n
a
0
+
a
1
+
...
+
a
n
.
3
1
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Perfect Square
Find all integer triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
with
a
>
0
>
b
>
c
a>0>b>c
a
>
0
>
b
>
c
whose sum equal
0
0
0
such that the number
N
=
2017
−
a
3
b
−
b
3
c
−
c
3
a
N=2017-a^3b-b^3c-c^3a
N
=
2017
−
a
3
b
−
b
3
c
−
c
3
a
is a perfect square of an integer.
2
1
Hide problems
Triangles in the Plane Containing a Point
Let
A
A
A
be a point in the plane and
3
3
3
lines which pass through this point divide the plane in
6
6
6
regions. In each region there are
5
5
5
points. We know that no three of the
30
30
30
points existing in these regions are collinear. Prove that there exist at least
1000
1000
1000
triangles whose vertices are points of those regions such that
A
A
A
lies either in the interior or on the side of the triangle.
1
1
Hide problems
Prove Points are Concyclic
An acute triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
is inscribed in a circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
. The circle
c
1
(
A
,
A
C
)
c_1(A,AC)
c
1
(
A
,
A
C
)
intersects the circle
c
c
c
at point
D
D
D
and intersects
C
B
CB
CB
at
E
E
E
. If the line
A
E
AE
A
E
intersects
c
c
c
at
F
F
F
and
G
G
G
lies in
B
C
BC
BC
such that
E
B
=
B
G
EB=BG
EB
=
BG
, prove that
F
,
E
,
D
,
G
F,E,D,G
F
,
E
,
D
,
G
are concyclic.