MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2005 Vietnam National Olympiad
2005 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
Find the smallest n
Let
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1A_2A_3A_4A_5A_6A_7A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
be convex 8-gon (no three diagonals concruent). The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1A_2A_3A_4A_5A_6A_7A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest
n
n
n
satisfying: We can color n "button" such that for all
i
,
k
∈
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
,
i
≠
k
,
s
(
i
,
k
)
i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)
i
,
k
∈
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
,
i
=
k
,
s
(
i
,
k
)
are the same where
s
(
i
,
k
)
s(i,k)
s
(
i
,
k
)
denote the number of the "sub quadrilaterals" has
A
i
,
A
k
A_i,A_k
A
i
,
A
k
be the vertices and the intersection of two its diagonals is "button".
x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n.Find the limit
Let
{
x
n
}
\{x_n\}
{
x
n
}
be a real sequence defined by:
x
1
=
a
,
x
n
+
1
=
3
x
n
3
−
7
x
n
2
+
5
x
n
x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n
x
1
=
a
,
x
n
+
1
=
3
x
n
3
−
7
x
n
2
+
5
x
n
For all
n
=
1
,
2
,
3...
n=1,2,3...
n
=
1
,
2
,
3...
and a is a real number. Find all
a
a
a
such that
{
x
n
}
\{x_n\}
{
x
n
}
has finite limit when
n
→
+
∞
n\to +\infty
n
→
+
∞
and find the finite limit in that cases.
2
2
Hide problems
x!+y!/n!=3^n
Find all triples of natural
(
x
,
y
,
n
)
(x,y,n)
(
x
,
y
,
n
)
satisfying the condition: \frac {x! \plus{} y!}{n!} \equal{} 3^n Define 0! \equal{} 1
A geometry problem from VMO 2005
Let
(
O
)
(O)
(
O
)
be a fixed circle with the radius
R
R
R
. Let
A
A
A
and
B
B
B
be fixed points in
(
O
)
(O)
(
O
)
such that
A
,
B
,
O
A,B,O
A
,
B
,
O
are not collinear. Consider a variable point
C
C
C
lying on
(
O
)
(O)
(
O
)
(
C
≠
A
,
B
C\neq A,B
C
=
A
,
B
). Construct two circles
(
O
1
)
,
(
O
2
)
(O_1),(O_2)
(
O
1
)
,
(
O
2
)
passing through
A
,
B
A,B
A
,
B
and tangent to
B
C
,
A
C
BC,AC
BC
,
A
C
at
C
C
C
, respectively. The circle
(
O
1
)
(O_1)
(
O
1
)
intersects the circle
(
O
2
)
(O_2)
(
O
2
)
in
D
D
D
(
D
≠
C
D\neq C
D
=
C
). Prove that: a)
C
D
≤
R
CD\leq R
C
D
≤
R
b) The line
C
D
CD
C
D
passes through a point independent of
C
C
C
(i.e. there exists a fixed point on the line
C
D
CD
C
D
when
C
C
C
lies on
(
O
)
(O)
(
O
)
).
1
2
Hide problems
Find the greatest value and the smallest value of x+y
Let
x
,
y
x,y
x
,
y
be real numbers satisfying the condition:
x
−
3
x
+
1
=
3
y
+
2
−
y
x-3\sqrt {x+1}=3\sqrt{y+2} -y
x
−
3
x
+
1
=
3
y
+
2
−
y
Find the greatest value and the smallest value of:
P
=
x
+
y
P=x+y
P
=
x
+
y
f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy
Find all function
f
:
R
→
R
f: \mathbb R\to \mathbb R
f
:
R
→
R
satisfying the condition: f(f(x \minus{} y)) \equal{} f(x)\cdot f(y) \minus{} f(x) \plus{} f(y) \minus{} xy