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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2004 Vietnam National Olympiad
2004 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
1
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Vietnam NMO 2004_6
Let
S
(
n
)
S(n)
S
(
n
)
be the sum of decimal digits of a natural number
n
n
n
. Find the least value of
S
(
m
)
S(m)
S
(
m
)
if
m
m
m
is an integral multiple of
2003
2003
2003
.
1
2
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Vietnam NMO 2004_1
Solve the system of equations \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}.
Vietnam NMO 2004_4
The sequence (x_n)^{\infty}_{n\equal{}1} is defined by x_1 \equal{} 1 and x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}, for all
n
∈
N
n \in\mathbb{N}
n
∈
N
, where
α
\alpha
α
is a given real parameter. Find all values of
α
\alpha
α
for which the sequence
(
y
n
)
(y_n)
(
y
n
)
given by y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1} has a finite limit when n \to \plus{}\infty and find that limit.
2
2
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Vietnam NMO 2004_2
In a triangle
A
B
C
ABC
A
BC
, the bisector of
∠
A
C
B
\angle ACB
∠
A
CB
cuts the side
A
B
AB
A
B
at
D
D
D
. An arbitrary circle
(
O
)
(O)
(
O
)
passing through
C
C
C
and
D
D
D
meets the lines
B
C
BC
BC
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
(different from
C
C
C
), respectively. (a) Prove that there is a circle
(
S
)
(S)
(
S
)
touching
D
M
DM
D
M
at
M
M
M
and
D
N
DN
D
N
at
N
N
N
. (b) If circle
(
S
)
(S)
(
S
)
intersects the lines
B
C
BC
BC
and
C
A
CA
C
A
again at
P
P
P
and
Q
Q
Q
respectively, prove that the lengths of the segments
M
P
MP
MP
and
N
Q
NQ
NQ
are constant as
(
O
)
(O)
(
O
)
varies.
Very interesting
Let
x
x
x
,
y
y
y
,
z
z
z
be positive reals satisfying
(
x
+
y
+
z
)
3
=
32
x
y
z
\left(x+y+z\right)^{3}=32xyz
(
x
+
y
+
z
)
3
=
32
x
yz
Find the minimum and the maximum of
P
=
x
4
+
y
4
+
z
4
(
x
+
y
+
z
)
4
P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}
P
=
(
x
+
y
+
z
)
4
x
4
+
y
4
+
z
4