MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2001 Vietnam National Olympiad
2001 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
x_{n+1} = x_n + b \sin x_n
For real
a
,
b
a, b
a
,
b
define the sequence
x
0
,
x
1
,
x
2
,
.
.
.
x_{0}, x_{1}, x_{2}, ...
x
0
,
x
1
,
x
2
,
...
by
x
0
=
a
,
x
n
+
1
=
x
n
+
b
sin
x
n
x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}
x
0
=
a
,
x
n
+
1
=
x
n
+
b
sin
x
n
. If
b
=
1
b = 1
b
=
1
, show that the sequence converges to a finite limit for all
a
a
a
. If
b
>
2
b > 2
b
>
2
, show that the sequence diverges for some
a
a
a
.
permutation of $\{1, 2, ... , 2n\}$
(
a
1
,
a
2
,
.
.
.
,
a
2
n
)
(a_{1}, a_{2}, ... , a_{2n})
(
a
1
,
a
2
,
...
,
a
2
n
)
is a permutation of
{
1
,
2
,
.
.
.
,
2
n
}
\{1, 2, ... , 2n\}
{
1
,
2
,
...
,
2
n
}
such that
∣
a
i
−
a
i
+
1
∣
≠
∣
a
j
−
a
j
+
1
∣
|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|
∣
a
i
−
a
i
+
1
∣
=
∣
a
j
−
a
j
+
1
∣
for
i
≠
j
i \neq j
i
=
j
. Show that
a
1
=
a
2
n
+
n
a_{1}= a_{2n}+n
a
1
=
a
2
n
+
n
iff
1
≤
a
2
i
≤
n
1 \leq a_{2i}\leq n
1
≤
a
2
i
≤
n
for
i
=
1
,
2
,
.
.
.
n
.
i = 1, 2, ... n.
i
=
1
,
2
,
...
n
.
2
2
Hide problems
Find the residue of $p^N+q^N \mod 6\cdot 12^n$.
Let
N
=
6
n
N = 6^{n}
N
=
6
n
, where
n
n
n
is a positive integer, and let
M
=
a
N
+
b
N
M = a^{N}+b^{N}
M
=
a
N
+
b
N
, where
a
a
a
and
b
b
b
are relatively prime integers greater than
1.
M
1. M
1.
M
has at least two odd divisors greater than
1
1
1
are
p
,
q
p,q
p
,
q
. Find the residue of
p
N
+
q
N
m
o
d
6
⋅
1
2
n
p^{N}+q^{N}\mod 6\cdot 12^{n}
p
N
+
q
N
mod
6
⋅
1
2
n
.
(1 - x^2) f(\frac{2x}{1 + x^2} ) = (1 + x^2)^2 f(x)
Find all real-valued continuous functions defined on the interval
(
−
1
,
1
)
(-1, 1)
(
−
1
,
1
)
such that
(
1
−
x
2
)
f
(
2
x
1
+
x
2
)
=
(
1
+
x
2
)
2
f
(
x
)
(1-x^{2}) f(\frac{2x}{1+x^{2}}) = (1+x^{2})^{2}f(x)
(
1
−
x
2
)
f
(
1
+
x
2
2
x
)
=
(
1
+
x
2
)
2
f
(
x
)
for all
x
x
x
.
1
2
Hide problems
$R, B$ and $R'$ are collinear
A circle center
O
O
O
meets a circle center
O
′
O'
O
′
at
A
A
A
and
B
.
B.
B
.
The line
T
T
′
TT'
T
T
′
touches the first circle at
T
T
T
and the second at
T
′
T'
T
′
. The perpendiculars from
T
T
T
and
T
′
T'
T
′
meet the line
O
O
′
OO'
O
O
′
at
S
S
S
and
S
′
S'
S
′
. The ray
A
S
AS
A
S
meets the first circle again at
R
R
R
, and the ray
A
S
′
AS'
A
S
′
meets the second circle again at
R
′
R'
R
′
. Show that
R
,
B
R, B
R
,
B
and
R
′
R'
R
′
are collinear.
Find the maximum value
Find the maximum value of
1
x
2
+
2
y
2
+
3
z
2
\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}
x
2
1
+
y
2
2
+
z
2
3
, where
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive reals satisfying
1
2
≤
z
<
min
(
x
2
,
y
3
)
2
,
x
+
z
3
≥
6
,
y
3
+
z
10
≥
2
5
.
\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.
2
1
≤
z
<
2
m
i
n
(
x
2
,
y
3
)
,
x
+
z
3
≥
6
,
y
3
+
z
10
≥
2
5
.