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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2016 Greece National Olympiad
2016 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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"Nice" Rhombuses In A Grid
A square
A
B
C
D
ABCD
A
BC
D
is divided into
n
2
n^2
n
2
equal small (fundamental) squares by drawing lines parallel to its sides.The vertices of the fundamental squares are called vertices of the grid.A rhombus is called nice when:
∙
\bullet
∙
It is not a square
∙
\bullet
∙
Its vertices are points of the grid
∙
\bullet
∙
Its diagonals are parallel to the sides of the square
A
B
C
D
ABCD
A
BC
D
Find (as a function of
n
n
n
) the number of the nice rhombuses (
n
n
n
is a positive integer greater than
2
2
2
).
3
1
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Greek contest 2016
A
B
C
ABC
A
BC
is an acute isosceles triangle
(
A
B
=
A
C
)
(AB=AC)
(
A
B
=
A
C
)
and
C
D
CD
C
D
one altitude. Circle
C
2
(
C
,
C
D
)
C_2(C,CD)
C
2
(
C
,
C
D
)
meets
A
C
AC
A
C
at
K
K
K
,
A
C
AC
A
C
produced at
Z
Z
Z
and circle
C
1
(
B
,
B
D
)
C_1(B, BD)
C
1
(
B
,
B
D
)
at
E
E
E
.
D
Z
DZ
D
Z
meets circle
(
C
1
)
(C_1)
(
C
1
)
at
M
M
M
. Show that: a)
Z
D
E
^
=
4
5
0
\widehat{ZDE}=45^0
Z
D
E
=
4
5
0
b) Points
E
,
M
,
K
E, M, K
E
,
M
,
K
lie on a line. c)
B
M
/
/
E
C
BM//EC
BM
//
EC
2
1
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Polynomials
Find all monic polynomials
P
,
Q
P,Q
P
,
Q
which are non-constant, have real coefficients and they satisfy
2
P
(
x
)
=
Q
(
(
x
+
1
)
2
2
)
−
Q
(
(
x
−
1
)
2
2
)
2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})
2
P
(
x
)
=
Q
(
2
(
x
+
1
)
2
)
−
Q
(
2
(
x
−
1
)
2
)
and
P
(
1
)
=
1
P(1)=1
P
(
1
)
=
1
for all real
x
x
x
.
1
1
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Diophantine equation
Find all triplets of nonnegative integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
and
x
≤
y
x\leq y
x
≤
y
such that
x
2
+
y
2
=
3
⋅
201
6
z
+
77
x^2+y^2=3 \cdot 2016^z+77
x
2
+
y
2
=
3
⋅
201
6
z
+
77