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Problems
Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2014 Greece Team Selection Test
2014 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(4)
4
1
Hide problems
Spider in square
Square
A
B
C
D
ABCD
A
BC
D
is divided into
n
2
n^2
n
2
equal small squares by lines parallel to its sides.A spider starts from
A
A
A
and moving only rightward or upwards,tries to reach
C
C
C
.Every "movement" of the spider consists of
k
k
k
steps rightward and
m
m
m
steps upwards or
m
m
m
steps rightward and
k
k
k
steps upwards(it can follow any possible order for the steps of each "movement").The spider completes
l
l
l
"movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If
n
=
m
⋅
l
n=m\cdot l
n
=
m
⋅
l
,find the number of the possible paths the spider can follow to reach
C
C
C
.Note that
n
,
m
,
k
,
l
∈
N
∗
n,m,k,l\in \mathbb{N^{*}}
n
,
m
,
k
,
l
∈
N
∗
with
k
<
m
k<m
k
<
m
.
3
1
Hide problems
Parallel lines
Let
A
B
C
ABC
A
BC
be an acute,non-isosceles triangle with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
.Let
D
,
E
,
Z
D,E,Z
D
,
E
,
Z
be the midpoints of
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively and segments
B
K
,
C
L
BK,CL
B
K
,
C
L
are altitudes.In the extension of
D
Z
DZ
D
Z
we take a point
M
M
M
such that the parallel from
M
M
M
to
K
L
KL
K
L
crosses the extensions of
C
A
,
B
A
,
D
E
CA,BA,DE
C
A
,
B
A
,
D
E
at
S
,
T
,
N
S,T,N
S
,
T
,
N
respectively (we extend
C
A
CA
C
A
to
A
A
A
-side and
B
A
BA
B
A
to
A
A
A
-side and
D
E
DE
D
E
to
E
E
E
-side).If the circumcirle
(
c
1
)
(c_{1})
(
c
1
)
of
△
M
B
D
\triangle{MBD}
△
MB
D
crosses the line
D
N
DN
D
N
at
R
R
R
and the circumcirle
(
c
2
)
(c_{2})
(
c
2
)
of
△
N
C
D
\triangle{NCD}
△
NC
D
crosses the line
D
M
DM
D
M
at
P
P
P
prove that
S
T
∥
P
R
ST\parallel PR
ST
∥
PR
.
1
1
Hide problems
Sequence with divisibility
Let
(
x
n
)
n
≥
1
(x_{n}) \ n\geq 1
(
x
n
)
n
≥
1
be a sequence of real numbers with
x
1
=
1
x_{1}=1
x
1
=
1
satisfying
2
x
n
+
1
=
3
x
n
+
5
x
n
2
−
4
2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}
2
x
n
+
1
=
3
x
n
+
5
x
n
2
−
4
a) Prove that the sequence consists only of natural numbers. b) Check if there are terms of the sequence divisible by
2011
2011
2011
.
2
1
Hide problems
Polynomial problem
Find all real non-zero polynomials satisfying
P
(
x
)
3
+
3
P
(
x
)
2
=
P
(
x
3
)
−
3
P
(
−
x
)
P(x)^3+3P(x)^2=P(x^{3})-3P(-x)
P
(
x
)
3
+
3
P
(
x
)
2
=
P
(
x
3
)
−
3
P
(
−
x
)
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
.