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Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2014 Vietnam Team Selection Test
2014 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
6
1
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Dien Bien fillings with integers of unit cubes, VNTST 2014 p6
m
,
n
,
p
m,n,p
m
,
n
,
p
are positive integers which do not simultaneously equal to zero.
3
3
3
D Cartesian space is divided into unit cubes by planes each perpendicular to one of
3
3
3
axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from
1
1
1
to
60
60
60
. A filling of integers is called Dien Bien if, for each rectangular box of size
{
2
m
+
1
,
2
n
+
1
,
2
p
+
1
}
\{2m+1,2n+1,2p+1\}
{
2
m
+
1
,
2
n
+
1
,
2
p
+
1
}
, the number in the unit cube which has common centre with the rectangular box is the average of the
8
8
8
numbers of the
8
8
8
unit cubes at the
8
8
8
corners of that rectangular box. How many Dien Bien fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation.translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here
2
1
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Vietnam TST 2014 - Problem 2
In the Cartesian plane is given a set of points with integer coordinate
T
=
{
(
x
;
y
)
∣
x
,
y
∈
Z
;
∣
x
∣
,
∣
y
∣
≤
20
;
(
x
;
y
)
≠
(
0
;
0
)
}
T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\}
T
=
{(
x
;
y
)
∣
x
,
y
∈
Z
;
∣
x
∣
,
∣
y
∣
≤
20
;
(
x
;
y
)
=
(
0
;
0
)}
We colour some points of
T
T
T
such that for each point
(
x
;
y
)
∈
T
(x;y)\in T
(
x
;
y
)
∈
T
then either
(
x
;
y
)
(x;y)
(
x
;
y
)
or
(
−
x
;
−
y
)
(-x;-y)
(
−
x
;
−
y
)
is coloured. Denote
N
N
N
to be the number of couples
(
x
1
;
y
1
)
,
(
x
2
;
y
2
)
{(x_1;y_1),(x_2;y_2)}
(
x
1
;
y
1
)
,
(
x
2
;
y
2
)
such that both
(
x
1
;
y
1
)
(x_1;y_1)
(
x
1
;
y
1
)
and
(
x
2
;
y
2
)
(x_2;y_2)
(
x
2
;
y
2
)
are coloured and
x
1
≡
2
x
2
(
m
o
d
41
)
,
y
1
≡
2
y
2
(
m
o
d
41
)
x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41}
x
1
≡
2
x
2
(
mod
41
)
,
y
1
≡
2
y
2
(
mod
41
)
. Find the all possible values of
N
N
N
.
1
1
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Functional equation
Find all
f
:
Z
→
Z
f:\mathbb{Z}\rightarrow\mathbb{Z}
f
:
Z
→
Z
such that
f
(
2
m
+
f
(
m
)
+
f
(
m
)
f
(
n
)
)
=
n
f
(
m
)
+
m
f(2m+f(m)+f(m)f(n))=nf(m)+m
f
(
2
m
+
f
(
m
)
+
f
(
m
)
f
(
n
))
=
n
f
(
m
)
+
m
∀
m
,
n
∈
Z
\forall m,n\in\mathbb{Z}
∀
m
,
n
∈
Z
5
1
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Polynomial
Find all polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
which have integer coefficients and satify the following condtion: For the sequence
(
x
n
)
(x_n )
(
x
n
)
defined by x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) n\geq 1 for every positive integer
m
m
m
is a divisor of some non-zero element of
(
x
n
)
(x_n )
(
x
n
)
3
1
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Viet Nam TST 2014 day 1 problem 3
Let
A
B
C
ABC
A
BC
be triangle with
A
<
B
<
C
A<B<C
A
<
B
<
C
and inscribed in a circle
(
O
)
(O)
(
O
)
. On the minor arc
A
B
C
ABC
A
BC
of
(
O
)
(O)
(
O
)
and does not contain point
A
A
A
, choose an arbitrary point
D
D
D
. Suppose
C
D
CD
C
D
meets
A
B
AB
A
B
at
E
E
E
and
B
D
BD
B
D
meets
A
C
AC
A
C
at
F
F
F
. Let
O
1
O_1
O
1
be the incenter of triangle
E
B
D
EBD
EB
D
touches with
E
B
,
E
D
EB,ED
EB
,
E
D
and tangent to
(
O
)
(O)
(
O
)
. Let
O
2
O_2
O
2
be the incenter of triangle
F
C
D
FCD
FC
D
, touches with
F
C
,
F
D
FC,FD
FC
,
F
D
and tangent to
(
O
)
(O)
(
O
)
. a)
M
M
M
is a tangency point of
O
1
O_1
O
1
with
B
E
BE
BE
and
N
N
N
is a tangency point of
O
2
O_2
O
2
with
C
F
CF
CF
. Prove that the circle with diameter
M
N
MN
MN
has a fixed point. b) A line through
M
M
M
is parallel to
C
E
CE
CE
meets
A
C
AC
A
C
at
P
P
P
, a line through
N
N
N
is parallel to
B
F
BF
BF
meets
A
B
AB
A
B
at
Q
Q
Q
. Prove that the circumcircles of triangles
(
A
M
P
)
,
(
A
N
Q
)
(AMP),(ANQ)
(
A
MP
)
,
(
A
NQ
)
are all tangent to a fixed circle.
4
1
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Viet Nam TST 2014 day 2 problem 1
a. Let
A
B
C
ABC
A
BC
be a triangle with altitude
A
D
AD
A
D
and
P
P
P
a variable point on
A
D
AD
A
D
. Lines
P
B
PB
PB
and
A
C
AC
A
C
intersect each other at
E
E
E
, lines
P
C
PC
PC
and
A
B
AB
A
B
intersect each other at
F
.
F.
F
.
Suppose
A
E
D
F
AEDF
A
E
D
F
is a quadrilateral inscribed . Prove that
P
A
P
D
=
(
tan
B
+
tan
C
)
cot
A
2
.
\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.
P
D
P
A
=
(
tan
B
+
tan
C
)
cot
2
A
.
b. Let
A
B
C
ABC
A
BC
be a triangle with orthocentre
H
H
H
and
P
P
P
a variable point on
A
H
AH
A
H
. The line through
C
C
C
perpendicular to
A
C
AC
A
C
meets
B
P
BP
BP
at
M
M
M
, The line through
B
B
B
perpendicular to
A
B
AB
A
B
meets
C
P
CP
CP
at
N
.
N.
N
.
K
K
K
is the projection of
A
A
A
on
M
N
MN
MN
. Prove that
∠
B
K
C
+
∠
M
A
N
\angle BKC+\angle MAN
∠
B
K
C
+
∠
M
A
N
is invariant .