MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2003 Vietnam National Olympiad
2003 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
number of permutations
Let
S
n
S_{n}
S
n
be the number of permutations
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_{1}, a_{2}, ... , a_{n})
(
a
1
,
a
2
,
...
,
a
n
)
of
(
1
,
2
,
.
.
.
,
n
)
(1, 2, ... , n)
(
1
,
2
,
...
,
n
)
such that
1
≤
∣
a
k
−
k
∣
≤
2
1 \leq |a_{k}-k | \leq 2
1
≤
∣
a
k
−
k
∣
≤
2
for all
k
k
k
. Show that
7
4
S
n
−
1
<
S
n
<
2
S
n
−
1
\frac{7}{4}S_{n-1}< S_{n}< 2 S_{n-1}
4
7
S
n
−
1
<
S
n
<
2
S
n
−
1
for
n
>
6.
n > 6.
n
>
6.
$f(x) \geq A x$
Let
F
\mathcal{F}
F
be the set of all functions
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f : (0,\infty)\to (0,\infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
such that
f
(
3
x
)
≥
f
(
f
(
2
x
)
)
+
x
f(3x) \geq f( f(2x) )+x
f
(
3
x
)
≥
f
(
f
(
2
x
))
+
x
for all
x
x
x
. Find the largest
A
A
A
such that
f
(
x
)
≥
A
x
f(x) \geq A x
f
(
x
)
≥
A
x
for all
f
∈
F
f\in\mathcal{F}
f
∈
F
and all
x
x
x
.
1
2
Hide problems
$g(x) = f(x) f(1-x)$
Let
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
is a function such that
f
(
cot
x
)
=
cos
2
x
+
sin
2
x
f( \cot x ) = \cos 2x+\sin 2x
f
(
cot
x
)
=
cos
2
x
+
sin
2
x
for all
0
<
x
<
π
0 < x < \pi
0
<
x
<
π
. Define
g
(
x
)
=
f
(
x
)
f
(
1
−
x
)
g(x) = f(x) f(1-x)
g
(
x
)
=
f
(
x
)
f
(
1
−
x
)
for
−
1
≤
x
≤
1
-1 \leq x \leq 1
−
1
≤
x
≤
1
. Find the maximum and minimum values of
g
g
g
on the closed interval
[
−
1
,
1
]
.
[-1, 1].
[
−
1
,
1
]
.
integer system (has root? )
Find the largest positive integer
n
n
n
such that the following equations have integer solutions in
x
,
y
1
,
y
2
,
.
.
.
,
y
n
x, y_{1}, y_{2}, ... , y_{n}
x
,
y
1
,
y
2
,
...
,
y
n
:
(
x
+
1
)
2
+
y
1
2
=
(
x
+
2
)
2
+
y
2
2
=
.
.
.
=
(
x
+
n
)
2
+
y
n
2
.
(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.
(
x
+
1
)
2
+
y
1
2
=
(
x
+
2
)
2
+
y
2
2
=
...
=
(
x
+
n
)
2
+
y
n
2
.
2
2
Hide problems
real roots of two plynomials
Define
p
(
x
)
=
4
x
3
−
2
x
2
−
15
x
+
9
,
q
(
x
)
=
12
x
3
+
6
x
2
−
7
x
+
1
p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1
p
(
x
)
=
4
x
3
−
2
x
2
−
15
x
+
9
,
q
(
x
)
=
12
x
3
+
6
x
2
−
7
x
+
1
. Show that each polynomial has just three distinct real roots. Let
A
A
A
be the largest root of
p
(
x
)
p(x)
p
(
x
)
and
B
B
B
the largest root of
q
(
x
)
q(x)
q
(
x
)
. Show that
A
2
+
3
B
2
=
4
A^{2}+3 B^{2}= 4
A
2
+
3
B
2
=
4
.
Straight line
The circles
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
touch externally at
M
M
M
and the radius of
C
2
C_{2}
C
2
is larger than that of
C
1
C_{1}
C
1
.
A
A
A
is any point on
C
2
C_{2}
C
2
which does not lie on the line joining the centers of the circles.
B
B
B
and
C
C
C
are points on
C
1
C_{1}
C
1
such that
A
B
AB
A
B
and
A
C
AC
A
C
are tangent to
C
1
C_{1}
C
1
. The lines
B
M
BM
BM
,
C
M
CM
CM
intersect
C
2
C_{2}
C
2
again at
E
E
E
,
F
F
F
respectively.
D
D
D
is the intersection of the tangent at
A
A
A
and the line
E
F
EF
EF
. Show that the locus of
D
D
D
as
A
A
A
varies is a straight line.