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Problems
Contests
National and Regional Contests
Vietnam Contests
Pre - Vietnam Mathematical Olympiad
2012 Pre - Vietnam Mathematical Olympiad
2012 Pre - Vietnam Mathematical Olympiad
Part of
Pre - Vietnam Mathematical Olympiad
Subcontests
(4)
4
1
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Game in grid
Two people A and B play a game in the
m
×
n
m \times n
m
×
n
grid (
m
,
n
∈
N
∗
m,n \in \mathbb{N^*}
m
,
n
∈
N
∗
). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?
3
2
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Triangle and circle
Let
A
B
C
ABC
A
BC
be a triangle with height
A
H
AH
A
H
.
P
P
P
lies on the circle over 3 midpoint of
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
(
P
∉
B
C
P \notin BC
P
∈
/
BC
). Prove that the line connect 2 center of
(
P
B
H
)
(PBH)
(
PB
H
)
and
(
P
C
H
)
(PCH)
(
PC
H
)
go through a fixed point. (where
(
X
Y
Z
)
(XYZ)
(
X
Y
Z
)
be a circumscribed circle of triangle
X
Y
Z
XYZ
X
Y
Z
)
Expand the capital
In a country, there are some cities and the city named Ben Song is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose
n
n
n
cities to add up with Ben Song city to establish an expanded capital such that the two following condition are satisfied:(i) With every two cities in expanded capital, we can always find a road connecting them and this road just belongs to the cities of expanded capital.(ii) There are exactly
k
k
k
cities which do not belong to expanded capital have the direct road to at least one city of expanded capital.Prove that there are at most
(
n
+
k
k
)
\binom{n+k}{k}
(
k
n
+
k
)
options to expand the capital for the King.
2
2
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Compute limit
Compute
lim
n
→
∞
{
(
2
+
3
)
n
}
\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}
n
→
∞
lim
{
(
2
+
3
)
n
}
Determine the first term of a sequence
Let
(
a
n
)
(a_n)
(
a
n
)
defined by:
a
0
=
1
,
a
1
=
p
,
a
2
=
p
(
p
−
1
)
a_0=1, \; a_1=p, \; a_2=p(p-1)
a
0
=
1
,
a
1
=
p
,
a
2
=
p
(
p
−
1
)
,
a
n
+
3
=
p
a
n
+
2
−
p
a
n
+
1
+
a
n
,
∀
n
∈
N
a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}
a
n
+
3
=
p
a
n
+
2
−
p
a
n
+
1
+
a
n
,
∀
n
∈
N
. Knowing that(i)
a
n
>
0
,
∀
n
∈
N
a_n>0, \; \forall n \in \mathbb{N}
a
n
>
0
,
∀
n
∈
N
.(ii)
a
m
a
n
>
a
m
+
1
a
n
−
1
,
∀
m
≥
n
≥
0
a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0
a
m
a
n
>
a
m
+
1
a
n
−
1
,
∀
m
≥
n
≥
0
.Prove that
∣
p
−
1
∣
≥
2
|p-1| \ge 2
∣
p
−
1∣
≥
2
.
1
2
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Simple Inequality
For
a
,
b
,
c
>
0
:
a
b
c
=
1
a,b,c>0: \; abc=1
a
,
b
,
c
>
0
:
ab
c
=
1
prove that
a
3
+
b
3
+
c
3
+
6
≥
(
a
+
b
+
c
)
2
a^3+b^3+c^3+6 \ge (a+b+c)^2
a
3
+
b
3
+
c
3
+
6
≥
(
a
+
b
+
c
)
2
2^n-1 | Sigma(2^n_i) => k>=n
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. Suppose there exist non-negative integers
n
1
,
n
2
,
…
,
n
k
{n_1},{n_2},\ldots,{n_k}
n
1
,
n
2
,
…
,
n
k
such that
2
n
−
1
∣
∑
i
=
1
k
2
n
i
2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}
2
n
−
1
∣
∑
i
=
1
k
2
n
i
. Prove that
k
≥
n
k \ge n
k
≥
n
.