MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2011 Vietnam National Olympiad
2011 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
Hide problems
There exists a unit circle - [VMO 2011]
A convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
satisfies that the sidelengths and
A
C
,
A
D
≤
3
.
AC,AD\leq \sqrt 3.
A
C
,
A
D
≤
3
.
Let us choose
2011
2011
2011
distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least
403
403
403
points out of the
2011
2011
2011
given points. {Edited} {I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}
2
2
Hide problems
Show that the sequence has finite limits - [VMO 2011]
Let
⟨
x
n
⟩
\langle x_n\rangle
⟨
x
n
⟩
be a sequence of real numbers defined as
x
1
=
1
;
x
n
=
2
n
(
n
−
1
)
2
∑
i
=
1
n
−
1
x
i
x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i
x
1
=
1
;
x
n
=
(
n
−
1
)
2
2
n
i
=
1
∑
n
−
1
x
i
Show that the sequence
y
n
=
x
n
+
1
−
x
n
y_n=x_{n+1}-x_n
y
n
=
x
n
+
1
−
x
n
has finite limits as
n
→
∞
.
n\to \infty.
n
→
∞.
A, M, N, P are concyclic iff d passes through I- [VMO 2011]
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
∠
C
\angle C
∠
C
and
∠
B
\angle B
∠
B
are acute. Let
D
D
D
be a variable point on
B
C
BC
BC
such that
D
≠
B
,
C
D\neq B, C
D
=
B
,
C
and
A
D
AD
A
D
is not perpendicular to
B
C
.
BC.
BC
.
Let
d
d
d
be the line passing through
D
D
D
and perpendicular to
B
C
.
BC.
BC
.
Assume
d
∩
A
B
=
E
,
d
∩
A
C
=
F
.
d \cap AB= E, d \cap AC =F.
d
∩
A
B
=
E
,
d
∩
A
C
=
F
.
If
M
,
N
,
P
M, N, P
M
,
N
,
P
are the incentres of
△
A
E
F
,
△
B
D
E
,
△
C
D
F
.
\triangle AEF, \triangle BDE,\triangle CDF.
△
A
EF
,
△
B
D
E
,
△
C
D
F
.
Prove that
A
,
M
,
N
,
P
A, M, N, P
A
,
M
,
N
,
P
are concyclic if and only if
d
d
d
passes through the incentre of
△
A
B
C
.
\triangle ABC.
△
A
BC
.
1
2
Hide problems
Prove the inequality for positive real x - [VMO 2011]
Prove that if
x
>
0
x>0
x
>
0
and
n
∈
N
,
n\in\mathbb N,
n
∈
N
,
then we have
x
n
(
x
n
+
1
+
1
)
x
n
+
1
≤
(
x
+
1
2
)
2
n
+
1
.
\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.
x
n
+
1
x
n
(
x
n
+
1
+
1
)
≤
(
2
x
+
1
)
2
n
+
1
.
Sequence; prove that 2011|[a_{2012}-2010]-[VMO 2011]
Define the sequence of integers
⟨
a
n
⟩
\langle a_n\rangle
⟨
a
n
⟩
as; a_0=1, a_1=-1, \text{ and } a_n=6a_{n-1}+5a_{n-2} \forall n\geq 2. Prove that
a
2012
−
2010
a_{2012}-2010
a
2012
−
2010
is divisible by
2011.
2011.
2011.
3
2
Hide problems
Concurrency problem and calculation - [VMO 2011]
Let
A
B
AB
A
B
be a diameter of a circle
(
O
)
(O)
(
O
)
and let
P
P
P
be any point on the tangent drawn at
B
B
B
to
(
O
)
.
(O).
(
O
)
.
Define
A
P
∩
(
O
)
=
C
≠
A
,
AP\cap (O)=C\neq A,
A
P
∩
(
O
)
=
C
=
A
,
and let
D
D
D
be the point diametrically opposite to
C
.
C.
C
.
If
D
P
DP
D
P
meets
(
O
)
(O)
(
O
)
second time in
E
,
E,
E
,
then,(i) Prove that
A
E
,
B
C
,
P
O
AE, BC, PO
A
E
,
BC
,
PO
concur at
M
.
M.
M
.
(ii) If
R
R
R
is the radius of
(
O
)
,
(O),
(
O
)
,
find
P
P
P
such that the area of
△
A
M
B
\triangle AMB
△
A
MB
is maximum, and calculate the area in terms of
R
.
R.
R
.
Prove that P(x,y) cannot be factorised -[VMO 2011]
Let
n
∈
N
n\in\mathbb N
n
∈
N
and define
P
(
x
,
y
)
=
x
n
+
x
y
+
y
n
.
P(x,y)=x^n+xy+y^n.
P
(
x
,
y
)
=
x
n
+
x
y
+
y
n
.
Show that we cannot obtain two non-constant polynomials
G
(
x
,
y
)
G(x,y)
G
(
x
,
y
)
and
H
(
x
,
y
)
H(x,y)
H
(
x
,
y
)
with real coefficients such that
P
(
x
,
y
)
=
G
(
x
,
y
)
⋅
H
(
x
,
y
)
.
P(x,y)=G(x,y)\cdot H(x,y).
P
(
x
,
y
)
=
G
(
x
,
y
)
⋅
H
(
x
,
y
)
.