MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1992 Vietnam National Olympiad
1992 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
2
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tetrahedron and its area surfaces
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron satisfying i)
A
C
D
^
+
B
C
D
^
=
18
0
0
\widehat{ACD}+\widehat{BCD}=180^{0}
A
C
D
+
BC
D
=
18
0
0
, and ii)
B
A
C
^
+
C
A
D
^
+
D
A
B
^
=
A
B
C
^
+
C
B
D
^
+
D
B
A
^
=
18
0
0
\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}
B
A
C
+
C
A
D
+
D
A
B
=
A
BC
+
CB
D
+
D
B
A
=
18
0
0
. Find value of
[
A
B
C
]
+
[
B
C
D
]
+
[
C
D
A
]
+
[
D
A
B
]
[ABC]+[BCD]+[CDA]+[DAB]
[
A
BC
]
+
[
BC
D
]
+
[
C
D
A
]
+
[
D
A
B
]
if we know
A
C
+
C
B
=
k
AC+CB=k
A
C
+
CB
=
k
and
A
C
B
^
=
α
\widehat{ACB}=\alpha
A
CB
=
α
.
root of polynomial
Let
9
<
n
1
<
n
2
<
…
<
n
s
<
1992
9 < n_{1} < n_{2} < \ldots < n_{s} < 1992
9
<
n
1
<
n
2
<
…
<
n
s
<
1992
be positive integers and P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}. Prove that if
x
0
x_{0}
x
0
is real root of
P
(
x
)
P(x)
P
(
x
)
then x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}.
2
2
Hide problems
last digit of divisor
For any positive integer
a
a
a
, denote
f
(
a
)
=
∣
{
b
∈
N
∣
b
∣
a
f(a)=|\{b\in\mathbb{N}| b|a
f
(
a
)
=
∣
{
b
∈
N
∣
b
∣
a
and
\text{and}
and
b
m
o
d
10
∈
{
1
,
9
}
}
∣
b\mod{10}\in\{1,9\}\}|
b
mod
10
∈
{
1
,
9
}}
∣
and
g
(
a
)
=
∣
{
b
∈
N
∣
b
∣
a
g(a)=|\{b\in\mathbb{N}| b|a
g
(
a
)
=
∣
{
b
∈
N
∣
b
∣
a
and
\text{and}
and
b
m
o
d
10
∈
{
3
,
7
}
}
∣
b\mod{10}\in\{3,7\}\}|
b
mod
10
∈
{
3
,
7
}}
∣
. Prove that
f
(
a
)
≥
g
(
a
)
∀
a
∈
N
f(a)\geq g(a)\forall a\in\mathbb{N}
f
(
a
)
≥
g
(
a
)
∀
a
∈
N
.
rotation rectangle
Let
H
H
H
be a rectangle with angle between two diagonal
≤
4
5
0
\leq 45^{0}
≤
4
5
0
. Rotation
H
H
H
around the its center with angle
0
0
≤
x
≤
36
0
0
0^{0}\leq x\leq 360^{0}
0
0
≤
x
≤
36
0
0
we have rectangle
H
x
H_{x}
H
x
. Find
x
x
x
such that
[
H
∩
H
x
]
[H\cap H_{x}]
[
H
∩
H
x
]
minimum, where
[
S
]
[S]
[
S
]
is area of
S
S
S
.
3
2
Hide problems
tend to infty
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals and sequences
{
a
n
}
,
{
b
n
}
,
{
c
n
}
\{a_{n}\},\{b_{n}\},\{c_{n}\}
{
a
n
}
,
{
b
n
}
,
{
c
n
}
defined by
a
k
+
1
=
a
k
+
2
b
k
+
c
k
,
b
k
+
1
=
b
k
+
2
c
k
+
a
k
,
c
k
+
1
=
c
k
+
2
a
k
+
b
k
a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}
a
k
+
1
=
a
k
+
b
k
+
c
k
2
,
b
k
+
1
=
b
k
+
c
k
+
a
k
2
,
c
k
+
1
=
c
k
+
a
k
+
b
k
2
for all
k
=
0
,
1
,
2
,
.
.
.
k=0,1,2,...
k
=
0
,
1
,
2
,
...
. Prove that
lim
k
→
+
∞
a
k
=
lim
k
→
+
∞
b
k
=
lim
k
→
+
∞
c
k
=
+
∞
\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty
lim
k
→
+
∞
a
k
=
lim
k
→
+
∞
b
k
=
lim
k
→
+
∞
c
k
=
+
∞
.
array of squares
Label the squares of a
1991
×
1992
1991 \times 1992
1991
×
1992
rectangle
(
m
,
n
)
(m, n)
(
m
,
n
)
with
1
≤
m
≤
1991
1 \leq m \leq 1991
1
≤
m
≤
1991
and
1
≤
n
≤
1992
1 \leq n \leq 1992
1
≤
n
≤
1992
. We wish to color all the squares red. The first move is to color red the squares
(
m
,
n
)
,
(
m
+
1
,
n
+
1
)
,
(
m
+
2
,
n
+
1
)
(m, n), (m+1, n+1), (m+2, n+1)
(
m
,
n
)
,
(
m
+
1
,
n
+
1
)
,
(
m
+
2
,
n
+
1
)
for some
m
<
1990
,
n
<
1992
m < 1990, n < 1992
m
<
1990
,
n
<
1992
. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?