MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1998 Vietnam National Olympiad
1998 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
a_{n+h}-a_n
The sequence
{
a
n
}
n
≥
0
\{a_{n}\}_{n\geq 0}
{
a
n
}
n
≥
0
is defined by
a
0
=
20
,
a
1
=
100
,
a
n
+
2
=
4
a
n
+
1
+
5
a
n
+
20
(
n
=
0
,
1
,
2
,
.
.
.
)
a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)
a
0
=
20
,
a
1
=
100
,
a
n
+
2
=
4
a
n
+
1
+
5
a
n
+
20
(
n
=
0
,
1
,
2
,
...
)
. Find the smallest positive integer
h
h
h
satisfying
1998
∣
a
n
+
h
−
a
n
∀
n
=
0
,
1
,
2
,
.
.
.
1998|a_{n+h}-a_{n}\forall n=0,1,2,...
1998∣
a
n
+
h
−
a
n
∀
n
=
0
,
1
,
2
,
...
polynomial
Find all positive integer
n
n
n
such that there exists a
P
∈
R
[
x
]
P\in\mathbb{R}[x]
P
∈
R
[
x
]
satisfying
P
(
x
1998
−
x
−
1998
)
=
x
n
−
x
−
n
∀
x
∈
R
−
{
0
}
P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}
P
(
x
1998
−
x
−
1998
)
=
x
n
−
x
−
n
∀
x
∈
R
−
{
0
}
.
2
2
Hide problems
tetrahedron and circumsphere
Let be given a tetrahedron whose circumcenter is
O
O
O
. Draw diameters
A
A
1
,
B
B
1
,
C
C
1
,
D
D
1
AA_{1},BB_{1},CC_{1},DD_{1}
A
A
1
,
B
B
1
,
C
C
1
,
D
D
1
of the circumsphere of
A
B
C
D
ABCD
A
BC
D
. Let
A
0
,
B
0
,
C
0
,
D
0
A_{0},B_{0},C_{0},D_{0}
A
0
,
B
0
,
C
0
,
D
0
be the centroids of triangle
B
C
D
,
C
D
A
,
D
A
B
,
A
B
C
BCD,CDA,DAB,ABC
BC
D
,
C
D
A
,
D
A
B
,
A
BC
. Prove that
A
0
A
1
,
B
0
B
1
,
C
0
C
1
,
D
0
D
1
A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}
A
0
A
1
,
B
0
B
1
,
C
0
C
1
,
D
0
D
1
are concurrent at a point, say,
F
F
F
. Prove that the line through
F
F
F
and a midpoint of a side of
A
B
C
D
ABCD
A
BC
D
is perpendicular to the opposite side.
find min
Find minimum value of
F
(
x
,
y
)
=
(
x
+
1
)
2
+
(
y
−
1
)
2
+
(
x
−
1
)
2
+
(
y
+
1
)
2
+
(
x
+
2
)
2
+
(
y
+
2
)
2
F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}
F
(
x
,
y
)
=
(
x
+
1
)
2
+
(
y
−
1
)
2
+
(
x
−
1
)
2
+
(
y
+
1
)
2
+
(
x
+
2
)
2
+
(
y
+
2
)
2
, where
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.
1
2
Hide problems
ln and sequence
Let
a
≥
1
a\geq 1
a
≥
1
be a real number. Put
x
1
=
a
,
x
n
+
1
=
1
+
ln
(
x
n
2
1
+
ln
x
n
)
(
n
=
1
,
2
,
.
.
.
)
x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)
x
1
=
a
,
x
n
+
1
=
1
+
ln
(
1
+
l
n
x
n
x
n
2
)
(
n
=
1
,
2
,
...
)
. Prove that the sequence
{
x
n
}
\{x_{n}\}
{
x
n
}
converges and find its limit.
Does there exist an infinite sequence ...?
Does there exist an infinite sequence
{
x
n
}
\{x_{n}\}
{
x
n
}
of reals satisfying the following conditions i)
∣
x
n
∣
≤
0
,
666
|x_{n}|\leq 0,666
∣
x
n
∣
≤
0
,
666
for all
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
ii)
∣
x
m
−
x
n
∣
≥
1
n
(
n
+
1
)
+
1
m
(
m
+
1
)
|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}
∣
x
m
−
x
n
∣
≥
n
(
n
+
1
)
1
+
m
(
m
+
1
)
1
for all
m
≠
n
m\not = n
m
=
n
?