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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2012 Greece National Olympiad
2012 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
1
1
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Positive integer couples
Let positive integers
p
,
q
p,q
p
,
q
with
gcd
(
p
,
q
)
=
1
\gcd(p,q)=1
g
cd
(
p
,
q
)
=
1
such as
p
+
q
2
=
(
n
2
+
1
)
p
2
+
q
p+q^2=(n^2+1)p^2+q
p
+
q
2
=
(
n
2
+
1
)
p
2
+
q
. If the parameter
n
n
n
is a positive integer, find all possible couples
(
p
,
q
)
(p,q)
(
p
,
q
)
.
2
1
Hide problems
Polynomials
Find all the non-zero polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
with real coefficients and the minimum degree,such that for all
x
∈
R
x \in \mathbb{R}
x
∈
R
:
P
(
x
2
)
+
Q
(
x
)
=
P
(
x
)
+
x
5
Q
(
x
)
P(x^2)+Q(x)=P(x)+x^5Q(x)
P
(
x
2
)
+
Q
(
x
)
=
P
(
x
)
+
x
5
Q
(
x
)
3
1
Hide problems
Concurrents and perpendicular bisector
Let an acute-angled triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
. The angle bisector
A
D
AD
A
D
meets
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at
K
K
K
. The circle
c
1
(
O
1
,
R
1
)
c_1(O_1,R_1)
c
1
(
O
1
,
R
1
)
(which passes from
A
,
D
A,D
A
,
D
and has its center
O
1
O_1
O
1
on
O
A
OA
O
A
) meets
A
B
AB
A
B
at
E
E
E
and
A
C
AC
A
C
at
Z
Z
Z
. If
M
,
N
M,N
M
,
N
are the midpoints of
Z
C
ZC
ZC
and
B
E
BE
BE
respectively, prove that: a)the lines
Z
E
,
D
M
,
K
C
ZE,DM,KC
ZE
,
D
M
,
K
C
are concurrent at one point
T
T
T
. b)the lines
Z
E
,
D
N
,
K
B
ZE,DN,KB
ZE
,
D
N
,
K
B
are concurrent at one point
X
X
X
. c)
O
K
OK
O
K
is the perpendicular bisector of
T
X
TX
TX
.
4
1
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Number of paths
The following isosceles trapezoid consists of equal equilateral triangles with side length
1
1
1
. The side
A
1
E
A_1E
A
1
E
has length
3
3
3
while the larger base
A
1
A
n
A_1A_n
A
1
A
n
has length
n
−
1
n-1
n
−
1
. Starting from the point
A
1
A_1
A
1
we move along the segments which are oriented to the right and up(obliquely right or left). Calculate (in terms of
n
n
n
or not) the number of all possible paths we can follow, in order to arrive at points
B
,
Γ
,
Δ
,
E
B,\Gamma,\Delta, E
B
,
Γ
,
Δ
,
E
, if
n
n
n
is an integer greater than
3
3
3
.[color=#00CCA7][Need image]