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Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2006 Vietnam Team Selection Test
2006 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
2
2
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Problem 2 vietnamese tst 2006
Find all pair of integer numbers
(
n
,
k
)
(n,k)
(
n
,
k
)
such that
n
n
n
is not negative and
k
k
k
is greater than
1
1
1
, and satisfying that the number:
A
=
1
7
2006
n
+
4.1
7
2
n
+
7.1
9
5
n
A=17^{2006n}+4.17^{2n}+7.19^{5n}
A
=
1
7
2006
n
+
4.1
7
2
n
+
7.1
9
5
n
can be represented as the product of
k
k
k
consecutive positive integers.
Problem 5 vietnamese tst 2006
Given a non-isoceles triangle
A
B
C
ABC
A
BC
inscribes a circle
(
O
,
R
)
(O,R)
(
O
,
R
)
(center
O
O
O
, radius
R
R
R
). Consider a varying line
l
l
l
such that
l
⊥
O
A
l\perp OA
l
⊥
O
A
and
l
l
l
always intersects the rays
A
B
,
A
C
AB,AC
A
B
,
A
C
and these intersectional points are called
M
,
N
M,N
M
,
N
. Suppose that the lines
B
N
BN
BN
and
C
M
CM
CM
intersect, and if the intersectional point is called
K
K
K
then the lines
A
K
AK
A
K
and
B
C
BC
BC
intersect.
1
1
1
, Assume that
P
P
P
is the intersectional point of
A
K
AK
A
K
and
B
C
BC
BC
. Show that the circumcircle of the triangle
M
N
P
MNP
MNP
is always through a fixed point.
2
2
2
, Assume that
H
H
H
is the orthocentre of the triangle
A
M
N
AMN
A
MN
. Denote
B
C
=
a
BC=a
BC
=
a
, and
d
d
d
is the distance between
A
A
A
and the line
H
K
HK
HK
. Prove that
d
≤
4
R
2
−
a
2
d\leq\sqrt{4R^2-a^2}
d
≤
4
R
2
−
a
2
and the equality occurs iff the line
l
l
l
is through the intersectional point of two lines
A
O
AO
A
O
and
B
C
BC
BC
.
1
2
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Problem 1 vietnamese tst 2006
Given an acute angles triangle
A
B
C
ABC
A
BC
, and
H
H
H
is its orthocentre. The external bisector of the angle
∠
B
H
C
\angle BHC
∠
B
H
C
meets the sides
A
B
AB
A
B
and
A
C
AC
A
C
at the points
D
D
D
and
E
E
E
respectively. The internal bisector of the angle
∠
B
A
C
\angle BAC
∠
B
A
C
meets the circumcircle of the triangle
A
D
E
ADE
A
D
E
again at the point
K
K
K
. Prove that
H
K
HK
HK
is through the midpoint of the side
B
C
BC
BC
.
Problem 4 vietnamese tst 2006
Prove that for all real numbers
x
,
y
,
z
∈
[
1
,
2
]
x,y,z \in [1,2]
x
,
y
,
z
∈
[
1
,
2
]
the following inequality always holds:
(
x
+
y
+
z
)
(
1
x
+
1
y
+
1
z
)
≥
6
(
x
y
+
z
+
y
z
+
x
+
z
x
+
y
)
.
(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}).
(
x
+
y
+
z
)
(
x
1
+
y
1
+
z
1
)
≥
6
(
y
+
z
x
+
z
+
x
y
+
x
+
y
z
)
.
When does the equality occur?
3
2
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Problem 3 vietnamese tst 2006
In the space are given
2006
2006
2006
distinct points, such that no
4
4
4
of them are coplanar. One draws a segment between each pair of points. A natural number
m
m
m
is called good if one can put on each of these segments a positive integer not larger than
m
m
m
, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a good number
m
m
m
.
Problem 6 vietnamese tst 2006
The real sequence
{
a
n
∣
n
=
0
,
1
,
2
,
3
,
.
.
.
}
\{a_n|n=0,1,2,3,...\}
{
a
n
∣
n
=
0
,
1
,
2
,
3
,
...
}
defined
a
0
=
1
a_0=1
a
0
=
1
and
a
n
+
1
=
1
2
(
a
n
+
1
3
⋅
a
n
)
.
a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ).
a
n
+
1
=
2
1
(
a
n
+
3
⋅
a
n
1
)
.
Denote
A
n
=
3
3
⋅
a
n
2
−
1
.
A_n=\frac{3}{3 \cdot a_n^2-1}.
A
n
=
3
⋅
a
n
2
−
1
3
.
Prove that
A
n
A_n
A
n
is a perfect square and it has at least
n
n
n
distinct prime divisors.