MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Pre - Vietnam Mathematical Olympiad
2011 Pre - Vietnam Mathematical Olympiad
2011 Pre - Vietnam Mathematical Olympiad
Part of
Pre - Vietnam Mathematical Olympiad
Subcontests
(4)
3
2
Hide problems
Two intersecting circles
Two circles
(
O
)
(O)
(
O
)
and
(
O
′
)
(O')
(
O
′
)
intersect at
A
A
A
and
B
B
B
. Take two points
P
,
Q
P,Q
P
,
Q
on
(
O
)
(O)
(
O
)
and
(
O
′
)
(O')
(
O
′
)
, respectively, such that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
. The line
P
Q
PQ
PQ
intersects
(
O
)
(O)
(
O
)
and
(
O
′
)
(O')
(
O
′
)
respectively at
M
,
N
M,N
M
,
N
. Let
E
,
F
E,F
E
,
F
respectively be the centers of the two arcs
B
P
BP
BP
and
B
Q
BQ
BQ
(which don't contains
A
A
A
). Prove that
M
N
E
F
MNEF
MNEF
is a cyclic quadrilateral.
Ineq of the selection in sets
There are
n
n
n
students. Denoted the number of the selections to select two students (with their weights are
a
a
a
and
b
b
b
) such that
∣
a
−
b
∣
≤
1
\left| {a - b} \right| \le 1
∣
a
−
b
∣
≤
1
(kg) and
∣
a
−
b
∣
≤
2
\left| {a - b} \right| \le 2
∣
a
−
b
∣
≤
2
(kg) by
A
1
A_1
A
1
and
A
2
A_2
A
2
, respectively. Prove that
A
2
<
3
A
1
+
n
A_2<3A_1+n
A
2
<
3
A
1
+
n
.
4
1
Hide problems
Problem involving table
For a table
n
×
9
n \times 9
n
×
9
(
n
n
n
rows and
9
9
9
columns), determine the maximum of
n
n
n
that we can write one number in the set
{
1
,
2
,
.
.
.
,
9
}
\left\{ {1,2,...,9} \right\}
{
1
,
2
,
...
,
9
}
in each cell such that these conditions are satisfied:1. Each row contains enough
9
9
9
numbers of the set
{
1
,
2
,
.
.
.
,
9
}
\left\{ {1,2,...,9} \right\}
{
1
,
2
,
...
,
9
}
.2. Any two rows are distinct.3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.
2
2
Hide problems
Ineq with the number of elements in the sets
Let
A
A
A
be a set of finite distinct positive real numbers. Two other sets
B
B
B
,
C
C
C
are defined by:
B
=
{
x
y
;
x
,
y
∈
A
}
,
C
=
{
x
y
;
x
,
y
∈
A
}
B = \left\{ {\frac{x}{y};x,y \in A} \right\},\; \; \; C = \left\{ {xy;x,y \in A} \right\}
B
=
{
y
x
;
x
,
y
∈
A
}
,
C
=
{
x
y
;
x
,
y
∈
A
}
Prove that
∣
A
∣
.
∣
B
∣
≤
∣
C
∣
2
\left| A \right|.\left| B \right| \le {\left| C \right|^2}
∣
A
∣
.
∣
B
∣
≤
∣
C
∣
2
.
System of functional equations
Find all function
f
,
g
:
Q
→
Q
f,g: \mathbb{Q} \to \mathbb{Q}
f
,
g
:
Q
→
Q
such that
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
1
2
Hide problems
Sequence
Let a sequence
{
x
n
}
\left\{ {{x_n}} \right\}
{
x
n
}
defined by:
{
x
0
=
−
2
x
n
=
1
−
1
−
4
x
n
−
1
2
,
∀
n
≥
1
\left\{ \begin{array}{l} {x_0} = - 2 \\ {x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\ \end{array} \right.
{
x
0
=
−
2
x
n
=
2
1
−
1
−
4
x
n
−
1
,
∀
n
≥
1
Denote
u
n
=
n
.
x
n
u_n=n.x_n
u
n
=
n
.
x
n
and
v
n
=
∏
i
=
0
n
(
1
+
x
i
2
)
{v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)}
v
n
=
i
=
0
∏
n
(
1
+
x
i
2
)
. Prove that
{
u
n
}
\left\{ {{u_n}} \right\}
{
u
n
}
,
{
v
n
}
\left\{ {{v_n}} \right\}
{
v
n
}
have finite limit.
Complete residue systems
Determine all values of
n
n
n
satisfied the following condition: there's exist a cyclic
(
a
1
,
a
2
,
a
3
,
.
.
.
,
a
n
)
(a_1,a_2,a_3,...,a_n)
(
a
1
,
a
2
,
a
3
,
...
,
a
n
)
of
(
1
,
2
,
3
,
.
.
.
,
n
)
(1,2,3,...,n)
(
1
,
2
,
3
,
...
,
n
)
such that
{
a
1
,
a
1
a
2
,
a
1
a
2
a
3
,
.
.
.
,
a
1
a
2
.
.
.
a
n
}
\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}
{
a
1
,
a
1
a
2
,
a
1
a
2
a
3
,
...
,
a
1
a
2
...
a
n
}
is a complete residue systems modulo
n
n
n
.