MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2015 Vietnam National Olympiad
2015 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
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VMO 2015 problem 4
Given a circumcircle
(
O
)
(O)
(
O
)
and two fixed points
B
,
C
B,C
B
,
C
on
(
O
)
(O)
(
O
)
.
B
C
BC
BC
is not the diameter of
(
O
)
(O)
(
O
)
. A point
A
A
A
varies on
(
O
)
(O)
(
O
)
such that
A
B
C
ABC
A
BC
is an acute triangle.
E
,
F
E,F
E
,
F
is the foot of the altitude from
B
,
C
B,C
B
,
C
respectively of
A
B
C
ABC
A
BC
.
(
I
)
(I)
(
I
)
is a variable circumcircle going through
E
E
E
and
F
F
F
with center
I
I
I
.a) Assume that
(
I
)
(I)
(
I
)
touches
B
C
BC
BC
at
D
D
D
. Probe that
D
B
D
C
=
cot
B
cot
C
\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}
D
C
D
B
=
c
o
t
C
c
o
t
B
.b) Assume
(
I
)
(I)
(
I
)
intersects
B
C
BC
BC
at
M
M
M
and
N
N
N
. Let
H
H
H
be the orthocenter and
P
,
Q
P,Q
P
,
Q
be the intersections of
(
I
)
(I)
(
I
)
and
(
H
B
C
)
(HBC)
(
H
BC
)
. The circumcircle
(
K
)
(K)
(
K
)
going through
P
,
Q
P,Q
P
,
Q
and touches
(
O
)
(O)
(
O
)
at
T
T
T
(
T
T
T
is on the same side with
A
A
A
wrt
P
Q
PQ
PQ
). Prove that the interior angle bisector of
∠
M
T
N
\angle{MTN}
∠
MTN
passes through a fixed point.
3
2
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VMO 2015 Problem 3
Given
m
∈
Z
+
m\in\mathbb{Z}^+
m
∈
Z
+
. Find all natural numbers
n
n
n
that does not exceed
1
0
m
10^m
1
0
m
satisfying the following conditions: i)
3
∣
n
.
3|n.
3∣
n
.
ii) The digits of
n
n
n
in decimal representation are in the set
{
2
,
0
,
1
,
5
}
\{2,0,1,5\}
{
2
,
0
,
1
,
5
}
.
VMO 2015 Problem 7
There are
m
m
m
boys and
n
n
n
girls participate in a duet singing contest (
m
,
n
≥
2
m,n\geq 2
m
,
n
≥
2
). At the contest, each section there will be one show. And a show including some boy-girl duets where each boy-girl couple will sing with each other no more than one song and each participant will sing at least one song. Two show
A
A
A
and
B
B
B
are considered different if there is a boy-girl couple sing at show
A
A
A
but not show
B
B
B
. The contest will end if and only if every possible shows are performed, and each show is performed exactly one time. a) A show is called depend on a participant
X
X
X
if we cancel all duets that
X
X
X
perform, then there will be at least one another participant will not sing any song in that show. Prove that among every shows that depend on
X
X
X
, the number of shows with odd number of songs equal to the number of shows with even number of songs.b) Prove that the organizers can arrange the shows in order to make sure that the numbers of songs in two consecutive shows have different parity.
1
2
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VMO 2015 Problem 1
Given a non negative real
a
a
a
and a sequence
(
u
n
)
(u_n)
(
u
n
)
defined by
{
u
1
=
3
u
n
+
1
=
u
n
2
+
n
2
4
n
2
+
a
u
n
2
+
3
\begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases}
{
u
1
=
3
u
n
+
1
=
2
u
n
+
4
n
2
+
a
n
2
u
n
2
+
3
a) Prove that for
a
=
0
a=0
a
=
0
, the sequence is convergent and find its limit.b) For
a
∈
[
0
,
1
]
a\in [0,1]
a
∈
[
0
,
1
]
, prove that the sequence if convergent.
Polynomial
Let
{
f
(
x
)
}
{\left\{ {f(x)} \right\}}
{
f
(
x
)
}
be a sequence of polynomial, where
f
0
(
x
)
=
2
{f_0}(x) = 2
f
0
(
x
)
=
2
,
f
1
(
x
)
=
3
x
{f_1}(x) = 3x
f
1
(
x
)
=
3
x
, and
f
n
(
x
)
=
3
x
f
n
−
1
(
x
)
+
(
1
−
x
−
2
x
2
)
f
n
−
2
(
x
)
{f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)
f
n
(
x
)
=
3
x
f
n
−
1
(
x
)
+
(
1
−
x
−
2
x
2
)
f
n
−
2
(
x
)
(
n
≥
2
)
(n \ge 2)
(
n
≥
2
)
Determine the value of
n
n
n
such that
f
n
(
x
)
{f_n}(x)
f
n
(
x
)
is divisible by
x
3
−
x
2
+
x
x^3-x^2+x
x
3
−
x
2
+
x
.
2
2
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Viet Nam National Math Olympiad 2015
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are nonnegative real numbers, then
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
a
b
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
.
{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
ab
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
.
VMO 2015 Problem 6
For
a
,
n
∈
Z
+
a,n\in\mathbb{Z}^+
a
,
n
∈
Z
+
, consider the following equation: a^2x+6ay+36z=n (1) where
x
,
y
,
z
∈
N
x,y,z\in\mathbb{N}
x
,
y
,
z
∈
N
.a) Find all
a
a
a
such that for all
n
≥
250
n\geq 250
n
≥
250
,
(
1
)
(1)
(
1
)
always has natural roots
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
.b) Given that
a
>
1
a>1
a
>
1
and
gcd
(
a
,
6
)
=
1
\gcd (a,6)=1
g
cd
(
a
,
6
)
=
1
. Find the greatest value of
n
n
n
in terms of
a
a
a
such that
(
1
)
(1)
(
1
)
doesn't have natural root
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
.