MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
2020 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(7)
7
1
Hide problems
Set and graph
Given a positive integer
n
>
1
n>1
n
>
1
. Denote
T
T
T
a set that contains all ordered sets
(
x
;
y
;
z
)
(x;y;z)
(
x
;
y
;
z
)
such that
x
,
y
,
z
x,y,z
x
,
y
,
z
are all distinct positive integers and
1
≤
x
,
y
,
z
≤
2
n
1\leq x,y,z\leq 2n
1
≤
x
,
y
,
z
≤
2
n
. Also, a set
A
A
A
containing ordered sets
(
u
;
v
)
(u;v)
(
u
;
v
)
is called "connected" with
T
T
T
if for every
(
x
;
y
;
z
)
∈
T
(x;y;z)\in T
(
x
;
y
;
z
)
∈
T
then
{
(
x
;
y
)
,
(
x
;
z
)
,
(
y
;
z
)
}
∩
A
≠
∅
\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing
{(
x
;
y
)
,
(
x
;
z
)
,
(
y
;
z
)}
∩
A
=
∅
. a) Find the number of elements of set
T
T
T
. b) Prove that there exists a set "connected" with
T
T
T
that has exactly
2
n
(
n
−
1
)
2n(n-1)
2
n
(
n
−
1
)
elements. c) Prove that every set "connected" with
T
T
T
has at least
2
n
(
n
−
1
)
2n(n-1)
2
n
(
n
−
1
)
elements.
6
1
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VMO 2020
Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively.
P
∈
A
B
,
Q
∈
A
C
P\in AB,Q\in AC
P
∈
A
B
,
Q
∈
A
C
satisfy
N
P
⊥
A
B
,
M
Q
⊥
A
C
NP\perp AB,MQ\perp AC
NP
⊥
A
B
,
MQ
⊥
A
C
a) Prove that EF is the tangent line of (APQ) b) Let T be the tangency point of (APQ) with EF,.DT
∩
\cap
∩
MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont
5
1
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VMO-D2-P5
Let a system of equations:
{
x
−
a
y
=
y
z
y
−
a
z
=
z
x
z
−
a
x
=
x
y
\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.
⎩
⎨
⎧
x
−
a
y
=
yz
y
−
a
z
=
z
x
z
−
a
x
=
x
y
a)Find (x,y,z) if a=0 b)Prove that: the system have 5 distinct roots
∀
\forall
∀
a>1,a
∈
R
.
\in\mathbb{R}.
∈
R
.
4
1
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VMO 2020 P4
Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a)
H
a
H_a
H
a
is the reflection of H in BC, A' is the reflection of A at O and
O
a
O_a
O
a
is the center of (BOC). Prove that
H
a
D
H_aD
H
a
D
and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A
3
1
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VMO 2020 -P3
Let a sequence
(
a
n
)
(a_n)
(
a
n
)
satisfy:
a
1
=
5
,
a
2
=
13
a_1=5,a_2=13
a
1
=
5
,
a
2
=
13
and
a
n
+
1
=
5
a
n
−
6
a
n
−
1
,
∀
n
≥
2
a_{n+1}=5a_n-6a_{n-1},\forall n\ge2
a
n
+
1
=
5
a
n
−
6
a
n
−
1
,
∀
n
≥
2
a) Prove that
(
a
n
,
a
n
+
1
)
=
1
,
∀
n
≥
1
(a_n, a_{n+1})=1,\forall n\ge1
(
a
n
,
a
n
+
1
)
=
1
,
∀
n
≥
1
b) Prove that:
2
k
+
1
∣
p
−
1
∀
k
∈
N
2^{k+1}|p-1\forall k\in\mathbb{N}
2
k
+
1
∣
p
−
1∀
k
∈
N
, if p is a prime factor of
a
2
k
a_{2^k}
a
2
k
2
1
Hide problems
Inequation
a)Let
a
,
b
,
c
∈
R
a,b,c\in\mathbb{R}
a
,
b
,
c
∈
R
and
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
.Prove that:
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
2
|a-b|+|b-c|+|c-a|\le2\sqrt{2}
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
2
b) Let
a
1
,
a
2
,
.
.
a
2019
∈
R
a_1,a_2,..a_{2019}\in\mathbb{R}
a
1
,
a
2
,
..
a
2019
∈
R
and
∑
i
=
1
2019
a
i
2
=
1
\sum_{i=1}^{2019}a_i^2=1
∑
i
=
1
2019
a
i
2
=
1
.Find the maximum of:
S
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
.
.
.
+
∣
a
2019
−
a
1
∣
S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|
S
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
...
+
∣
a
2019
−
a
1
∣
1
1
Hide problems
VMO-2020
Let a sequence
(
x
n
)
(x_n)
(
x
n
)
satisfy :
x
1
=
1
x_1=1
x
1
=
1
and
x
n
+
1
=
x
n
+
3
x
n
+
n
x
n
x_{n+1}=x_n+3\sqrt{x_n} + \frac{n}{\sqrt{x_n}}
x
n
+
1
=
x
n
+
3
x
n
+
x
n
n
,
∀
\forall
∀
n
≥
1
\ge1
≥
1
a) Prove lim
n
x
n
=
0
\frac{n}{x_n}=0
x
n
n
=
0
b) Find lim
n
2
x
n
\frac{n^2}{x_n}
x
n
n
2