MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2008 Vietnam Team Selection Test
2008 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
wanting set
Let an integer
n
>
3
n > 3
n
>
3
. Denote the set T\equal{}\{1,2, \ldots,n\}. A subset S of T is called wanting set if S has the property: There exists a positive integer
c
c
c
which is not greater than
n
2
\frac {n}{2}
2
n
such that |s_1 \minus{} s_2|\ne c for every pairs of arbitrary elements
s
1
,
s
2
∈
S
s_1,s_2\in S
s
1
,
s
2
∈
S
. How many does a wanting set have at most are there ?
two sets
Consider the set
M
=
{
1
,
2
,
…
,
2008
}
M = \{1,2, \ldots ,2008\}
M
=
{
1
,
2
,
…
,
2008
}
. Paint every number in the set
M
M
M
with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
S
1
=
{
(
x
,
y
,
z
)
∈
M
3
∣
x
,
y
,
z
have the same color and
2008
∣
(
x
+
y
+
z
)
}
S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}
S
1
=
{(
x
,
y
,
z
)
∈
M
3
∣
x
,
y
,
z
have the same color and
2008∣
(
x
+
y
+
z
)}
;
S
2
=
{
(
x
,
y
,
z
)
∈
M
3
∣
x
,
y
,
z
have three pairwisely different colors and
2008
∣
(
x
+
y
+
z
)
}
S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}
S
2
=
{(
x
,
y
,
z
)
∈
M
3
∣
x
,
y
,
z
have three pairwisely different colors and
2008∣
(
x
+
y
+
z
)}
. Prove that
2
∣
S
1
∣
>
∣
S
2
∣
2|S_1| > |S_2|
2∣
S
1
∣
>
∣
S
2
∣
(where
∣
X
∣
|X|
∣
X
∣
denotes the number of elements in a set
X
X
X
).
2
2
Hide problems
polynomials
Find all values of the positive integer
m
m
m
such that there exists polynomials
P
(
x
)
,
Q
(
x
)
,
R
(
x
,
y
)
P(x),Q(x),R(x,y)
P
(
x
)
,
Q
(
x
)
,
R
(
x
,
y
)
with real coefficient satisfying the condition: For every real numbers
a
,
b
a,b
a
,
b
which satisfying
a
m
−
b
2
=
0
a^m-b^2=0
a
m
−
b
2
=
0
, we always have that
P
(
R
(
a
,
b
)
)
=
a
P(R(a,b))=a
P
(
R
(
a
,
b
))
=
a
and
Q
(
R
(
a
,
b
)
)
=
b
Q(R(a,b))=b
Q
(
R
(
a
,
b
))
=
b
.
three circles
Let
k
k
k
be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k. Denote
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1),(O_2),(O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C. 1) Prove that when k \equal{} \frac{1}{2}, three circles
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1),(O_2),(O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles. 2) Find all values of k such that three circles
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1),(O_2),(O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
have exactly two common points
1
2
Hide problems
Prove lines are concurrent.
On the plane, given an angle
x
O
y
xOy
x
O
y
.
M
M
M
be a mobile point on ray
O
x
Ox
O
x
and
N
N
N
a mobile point on ray
O
y
Oy
O
y
. Let
d
d
d
be the external angle bisector of angle
x
O
y
xOy
x
O
y
and
I
I
I
be the intersection of
d
d
d
with the perpendicular bisector of
M
N
MN
MN
. Let
P
P
P
,
Q
Q
Q
be two points lie on
d
d
d
such that IP \equal{} IQ \equal{} IM \equal{} IN, and let
K
K
K
the intersection of
M
Q
MQ
MQ
and
N
P
NP
NP
.
1.
1.
1.
Prove that
K
K
K
always lie on a fixed line.
2.
2.
2.
Let
d
1
d_1
d
1
line perpendicular to
I
M
IM
I
M
at
M
M
M
and
d
2
d_2
d
2
line perpendicular to
I
N
IN
I
N
at
N
N
N
. Assume that there exist the intersections
E
E
E
,
F
F
F
of
d
1
d_1
d
1
,
d
2
d_2
d
2
from
d
d
d
. Prove that
E
N
EN
EN
,
F
M
FM
FM
and
O
K
OK
O
K
are concurrent.
6m|(2m + 3)^n+1 if and only if 4m|3^n+1
Let
m
m
m
and
n
n
n
be positive integers. Prove that 6m | (2m \plus{} 3)^n \plus{} 1 if and only if 4m | 3^n \plus{} 1