MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2006 Vietnam National Olympiad
2006 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(6)
6
1
Hide problems
Vmo 2006 a6
Let
S
S
S
be a set of 2006 numbers. We call a subset
T
T
T
of
S
S
S
naughty if for any two arbitrary numbers
u
u
u
,
v
v
v
(not neccesary distinct) in
T
T
T
,
u
+
v
u+v
u
+
v
is not in
T
T
T
. Prove that 1) If
S
=
{
1
,
2
,
…
,
2006
}
S=\{1,2,\ldots,2006\}
S
=
{
1
,
2
,
…
,
2006
}
every naughty subset of
S
S
S
has at most 1003 elements; 2) If
S
S
S
is a set of 2006 arbitrary positive integers, there exists a naughty subset of
S
S
S
which has 669 elements.
5
1
Hide problems
Vmo 2006 a5
Find all polynomyals
P
(
x
)
P(x)
P
(
x
)
with real coefficients which satisfy the following equality for all real numbers
x
x
x
:
P
(
x
2
)
+
x
(
3
P
(
x
)
+
P
(
−
x
)
)
=
(
P
(
x
)
)
2
+
2
x
2
.
P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 .
P
(
x
2
)
+
x
(
3
P
(
x
)
+
P
(
−
x
))
=
(
P
(
x
)
)
2
+
2
x
2
.
4
1
Hide problems
Vmo 2006 a4
Given is the function
f
(
x
)
=
−
x
+
(
x
+
a
)
(
x
+
b
)
f(x)=-x+\sqrt{(x+a)(x+b)}
f
(
x
)
=
−
x
+
(
x
+
a
)
(
x
+
b
)
, where
a
a
a
,
b
b
b
are distinct given positive real numbers. Prove that for all real numbers
s
∈
(
0
,
1
)
s\in (0,1)
s
∈
(
0
,
1
)
there exist only one positive real number
α
\alpha
α
such that
f
(
α
)
=
a
s
+
b
s
2
.
f(\alpha)=\sqrt {\frac{a^s+b^s}{2}} .
f
(
α
)
=
2
a
s
+
b
s
.
3
1
Hide problems
Vmo 2006 a3
Let
m
m
m
,
n
n
n
be two positive integers greater than 3. Consider the table of size
m
×
n
m\times n
m
×
n
(
m
m
m
rows and
n
n
n
columns) formed with unit squares. We are putting marbles into unit squares of the table following the instructions:
−
-
−
each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click [url=http://www.mathlinks.ro/Forum/download.php?id=4425]here to view the pictures). In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same? a)
m
=
2004
m=2004
m
=
2004
,
n
=
2006
n=2006
n
=
2006
; b)
m
=
2005
m=2005
m
=
2005
,
n
=
2006
n=2006
n
=
2006
.
2
1
Hide problems
Vmo 2006 a2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Take an arbitrary point
M
M
M
on the line
A
B
AB
A
B
, and let
N
N
N
be the point of intersection of the circumcircles of triangles
M
A
C
MAC
M
A
C
and
M
B
C
MBC
MBC
(different from
M
M
M
). Prove that: a) The point
N
N
N
lies on a fixed circle; b) The line
M
N
MN
MN
passes though a fixed point.
1
1
Hide problems
Vmo 2006 a1
Solve the following system of equations in real numbers:
{
x
2
−
2
x
+
6
⋅
log
3
(
6
−
y
)
=
x
y
2
−
2
y
+
6
⋅
log
3
(
6
−
z
)
=
y
z
2
−
2
z
+
6
⋅
log
3
(
6
−
x
)
=
z
.
\begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}.
⎩
⎨
⎧
x
2
−
2
x
+
6
⋅
lo
g
3
(
6
−
y
)
=
x
y
2
−
2
y
+
6
⋅
lo
g
3
(
6
−
z
)
=
y
z
2
−
2
z
+
6
⋅
lo
g
3
(
6
−
x
)
=
z
.