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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2007 Vietnam Team Selection Test
2007 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
2
1
Hide problems
triangle and circles
Let
A
B
C
ABC
A
BC
be an acute triangle with incricle
(
I
)
(I)
(
I
)
.
(
K
A
)
(K_{A})
(
K
A
)
is the cricle such that
A
∈
(
K
A
)
A\in (K_{A})
A
∈
(
K
A
)
and
A
K
A
⊥
B
C
AK_{A}\perp BC
A
K
A
⊥
BC
and it in-tangent for
(
I
)
(I)
(
I
)
at
A
1
A_{1}
A
1
, similary we have
B
1
,
C
1
B_{1},C_{1}
B
1
,
C
1
. a) Prove that
A
A
1
,
B
B
1
,
C
C
1
AA_{1},BB_{1},CC_{1}
A
A
1
,
B
B
1
,
C
C
1
are concurrent, called point-concurrent is
P
P
P
. b) Assume circles
(
J
A
)
,
(
J
B
)
,
(
J
C
)
(J_{A}),(J_{B}),(J_{C})
(
J
A
)
,
(
J
B
)
,
(
J
C
)
are symmetry for excircles
(
I
A
)
,
(
I
B
)
,
(
I
C
)
(I_{A}),(I_{B}),(I_{C})
(
I
A
)
,
(
I
B
)
,
(
I
C
)
across midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
,resp. Prove that
P
P
/
(
J
A
)
=
P
P
/
(
J
B
)
=
P
P
/
(
J
C
)
P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}
P
P
/
(
J
A
)
=
P
P
/
(
J
B
)
=
P
P
/
(
J
C
)
. Note. If
(
O
;
R
)
(O;R)
(
O
;
R
)
is a circle and
M
M
M
is a point then
P
M
/
(
O
)
=
O
M
2
−
R
2
P_{M/(O)}=OM^{2}-R^{2}
P
M
/
(
O
)
=
O
M
2
−
R
2
.
6
1
Hide problems
9-gon regular
Let
A
1
A
2
…
A
9
A_{1}A_{2}\ldots A_{9}
A
1
A
2
…
A
9
be a regular
9
−
9-
9
−
gon. Let
{
A
1
,
A
2
,
…
,
A
9
}
=
S
1
∪
S
2
∪
S
3
\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}
{
A
1
,
A
2
,
…
,
A
9
}
=
S
1
∪
S
2
∪
S
3
such that
∣
S
1
∣
=
∣
S
2
∣
=
∣
S
3
∣
=
3
|S_{1}|=|S_{2}|=|S_{3}|=3
∣
S
1
∣
=
∣
S
2
∣
=
∣
S
3
∣
=
3
. Prove that there exists
A
,
B
∈
S
1
A,B\in S_{1}
A
,
B
∈
S
1
,
C
,
D
∈
S
2
C,D\in S_{2}
C
,
D
∈
S
2
,
E
,
F
∈
S
3
E,F\in S_{3}
E
,
F
∈
S
3
such that
A
B
=
C
D
=
E
F
AB=CD=EF
A
B
=
C
D
=
EF
and
A
≠
B
A \neq B
A
=
B
,
C
≠
D
C\neq D
C
=
D
,
E
≠
F
E\neq F
E
=
F
.
5
1
Hide problems
$A\subset \{1,2,...,4014\}$
Let
A
⊂
{
1
,
2
,
…
,
4014
}
A\subset \{1,2,\ldots,4014\}
A
⊂
{
1
,
2
,
…
,
4014
}
,
∣
A
∣
=
2007
|A|=2007
∣
A
∣
=
2007
, such that
a
a
a
does not divide
b
b
b
for all distinct elements
a
,
b
∈
A
a,b\in A
a
,
b
∈
A
. For a set
X
X
X
as above let us denote with
m
X
m_{X}
m
X
the smallest element in
X
X
X
. Find
min
m
A
\min m_{A}
min
m
A
(for all
A
A
A
with the above properties).
4
1
Hide problems
find continuous function
Find all continuous functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that for all real
x
x
x
we have
f
(
x
)
=
f
(
x
2
+
x
3
+
1
9
)
.
f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right).
f
(
x
)
=
f
(
x
2
+
3
x
+
9
1
)
.
1
1
Hide problems
nxn array
Given two sets
A
,
B
A, B
A
,
B
of positive real numbers such that:
∣
A
∣
=
∣
B
∣
=
n
|A| = |B| =n
∣
A
∣
=
∣
B
∣
=
n
;
A
≠
B
A \neq B
A
=
B
and
S
(
A
)
=
S
(
B
)
S(A)=S(B)
S
(
A
)
=
S
(
B
)
, where
∣
X
∣
|X|
∣
X
∣
is the number of elements and
S
(
X
)
S(X)
S
(
X
)
is the sum of all elements in set
X
X
X
. Prove that we can fill in each unit square of a
n
×
n
n\times n
n
×
n
square with positive numbers and some zeros such that: a) the set of the sum of all numbers in each row equals
A
A
A
; b) the set of the sum of all numbers in each column equals
A
A
A
. c) there are at least
(
n
−
1
)
2
+
k
(n-1)^{2}+k
(
n
−
1
)
2
+
k
zero numbers in the
n
×
n
n\times n
n
×
n
array with
k
=
∣
A
∩
B
∣
k=|A \cap B|
k
=
∣
A
∩
B
∣
.
3
1
Hide problems
Vietnam TST 2007
Given a triangle
A
B
C
ABC
A
BC
. Find the minimum of
cos
2
A
2
cos
2
B
2
cos
2
C
2
+
cos
2
B
2
cos
2
C
2
cos
2
A
2
+
cos
2
C
2
cos
2
A
2
cos
2
B
2
.
\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}.
cos
2
2
C
cos
2
2
A
cos
2
2
B
+
cos
2
2
A
cos
2
2
B
cos
2
2
C
+
cos
2
2
B
cos
2
2
C
cos
2
2
A
.