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National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2010 Vietnam Team Selection Test
2010 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
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VN TST 2010 Pro 3
We call a rectangle of the size
1
×
2
1 \times 2
1
×
2
a domino. Rectangle of the
2
×
3
2 \times 3
2
×
3
removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.It requires to tile rectangle of size
2008
×
2010
2008 \times 2010
2008
×
2010
by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
VN TST 2010 Pro 6
Let
S
n
S_n
S
n
be sum of squares of the coefficient of the polynomial
(
1
+
x
)
n
(1+x)^n
(
1
+
x
)
n
. Prove that
S
2
n
+
1
S_{2n} +1
S
2
n
+
1
is not divisible by
3.
3.
3.
2
2
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VNTST 2010 Pro 2
Let
A
B
C
ABC
A
BC
be a triangle with
B
A
C
^
≠
9
0
∘
\widehat{BAC}\neq 90^\circ
B
A
C
=
9
0
∘
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. We choose a variable point
D
D
D
on
A
M
AM
A
M
. Let
(
O
1
)
(O_1)
(
O
1
)
and
(
O
2
)
(O_2)
(
O
2
)
be two circle pass through
D
D
D
and tangent to
B
C
BC
BC
at
B
B
B
and
C
C
C
. The line
B
A
BA
B
A
and
C
A
CA
C
A
intersect
(
O
1
)
,
(
O
2
)
(O_1),(O_2)
(
O
1
)
,
(
O
2
)
at
P
,
Q
P,Q
P
,
Q
respectively.a) Prove that tangent line at
P
P
P
on
(
O
1
)
(O_1)
(
O
1
)
and
Q
Q
Q
on
(
O
2
)
(O_2)
(
O
2
)
must intersect at
S
S
S
.b) Prove that
S
S
S
lies on a fix line.
VN TST 2010 Pro 5
We have
n
n
n
countries. Each country have
m
m
m
persons who live in that country (
n
>
m
>
1
n>m>1
n
>
m
>
1
). We divide
m
⋅
n
m \cdot n
m
⋅
n
persons into
n
n
n
groups each with
m
m
m
members such that there don't exist two persons in any groups who come from one country. Prove that one can choose
n
n
n
people into one class such that they come from different groups and different countries.
1
2
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Viet Nam TST 2010 Pro 1
Let
n
n
n
be a positive integer. Let
T
n
T_n
T
n
be a set of positive integers such that:
T
n
=
{
11
(
k
+
h
)
+
10
(
n
k
+
n
h
)
∣
(
1
≤
k
,
h
≤
10
)
}
{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}
T
n
=
{
11
(
k
+
h
)
+
10
(
n
k
+
n
h
)
∣
(
1
≤
k
,
h
≤
10
)
}
Find all
n
n
n
for which there don't exist two distinct positive integers
a
,
b
∈
T
n
a, b \in T_n
a
,
b
∈
T
n
such that
a
≡
b
(
m
o
d
110
)
a\equiv b \pmod{110}
a
≡
b
(
mod
110
)
VN TST 2010 Pro 4
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers which satisfy the condition:
16
(
a
+
b
+
c
)
≥
1
a
+
1
b
+
1
c
16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}
16
(
a
+
b
+
c
)
≥
a
1
+
b
1
+
c
1
. Prove that
∑
c
y
c
(
1
a
+
b
+
2
a
+
2
c
)
3
≤
8
9
\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}
cyc
∑
(
a
+
b
+
2
a
+
2
c
1
)
3
≤
9
8