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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2023 Vietnam National Olympiad
2023 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(7)
1
1
Hide problems
(a_n) and (b_n) have finite limits
Consider the sequence
(
a
n
)
(a_n)
(
a
n
)
satisfying
a
1
=
1
2
,
a
n
+
1
=
3
a
n
+
1
−
a
n
3
a_1=\dfrac{1}{2},a_{n+1}=\sqrt[3]{3a_{n+1}-a_n}
a
1
=
2
1
,
a
n
+
1
=
3
3
a
n
+
1
−
a
n
and
0
≤
a
n
≤
1
,
∀
n
≥
1.
0\le a_n\le 1,\forall n\ge 1.
0
≤
a
n
≤
1
,
∀
n
≥
1.
a. Prove that the sequence
(
a
n
)
(a_n)
(
a
n
)
is determined uniquely and has finite limit.b. Let
b
n
=
(
1
+
2.
a
1
)
(
1
+
2
2
a
2
)
.
.
.
(
1
+
2
n
a
n
)
,
∀
n
≥
1.
b_n=(1+2.a_1)(1+2^2a_2)...(1+2^na_n), \forall n\ge 1.
b
n
=
(
1
+
2.
a
1
)
(
1
+
2
2
a
2
)
...
(
1
+
2
n
a
n
)
,
∀
n
≥
1.
Prove that the sequence
(
b
n
)
(b_n)
(
b
n
)
has finite limit.
6
1
Hide problems
Set theory
There are
n
≥
2
n \geq 2
n
≥
2
classes organized
m
≥
1
m \geq 1
m
≥
1
extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly
a
a
a
classes that the students in this group participate in. For any two extracurricular groups, there are no more than
b
b
b
classes with students participating in both groups simultaneously. a) Find
m
m
m
when
n
=
8
,
a
=
4
,
b
=
1
n = 8, a = 4 , b = 1
n
=
8
,
a
=
4
,
b
=
1
. b) Prove that
n
≥
20
n \geq 20
n
≥
20
when
m
=
6
,
a
=
10
,
b
=
4
m = 6 , a = 10 , b = 4
m
=
6
,
a
=
10
,
b
=
4
. c) Find the minimum value of
n
n
n
when
m
=
20
,
a
=
4
,
b
=
1
m = 20 , a = 4 , b = 1
m
=
20
,
a
=
4
,
b
=
1
.
5
1
Hide problems
FE with two functions
Find all functions
f
,
g
:
R
→
R
f, g: \mathbb{R} \rightarrow \mathbb{R}
f
,
g
:
R
→
R
satisfying
f
(
0
)
=
2022
f (0)=2022
f
(
0
)
=
2022
and
f
(
x
+
g
(
y
)
)
=
x
f
(
y
)
+
(
2023
−
y
)
f
(
x
)
+
g
(
x
)
f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)
f
(
x
+
g
(
y
))
=
x
f
(
y
)
+
(
2023
−
y
)
f
(
x
)
+
g
(
x
)
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
7
1
Hide problems
Geometry with tangents and inequality
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be a scalene triangle with orthocenter
H
H
H
and circumcenter
O
O
O
. Incircle
(
I
)
(I)
(
I
)
of the
△
A
B
C
\triangle{ABC}
△
A
BC
is tangent to the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
M
,
N
,
P
M,N,P
M
,
N
,
P
respectively. Denote
Ω
A
\Omega_A
Ω
A
to be the circle passing through point
A
A
A
, external tangent to
(
I
)
(I)
(
I
)
at
A
′
A'
A
′
and cut again
A
B
,
A
C
AB,AC
A
B
,
A
C
at
A
b
,
A
c
A_b,A_c
A
b
,
A
c
respectively. The circles
Ω
B
,
Ω
C
\Omega_B,\Omega_C
Ω
B
,
Ω
C
and points
B
′
,
B
a
,
B
c
,
C
′
,
C
a
,
C
b
B',B_a,B_c,C',C_a,C_b
B
′
,
B
a
,
B
c
,
C
′
,
C
a
,
C
b
are defined similarly.
a
)
a)
a
)
Prove
B
c
C
b
+
C
a
A
c
+
A
b
B
a
≥
N
P
+
P
M
+
M
N
B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN
B
c
C
b
+
C
a
A
c
+
A
b
B
a
≥
NP
+
PM
+
MN
.
b
)
b)
b
)
Suppose
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
lie on
A
M
,
B
N
,
C
P
AM,BN,CP
A
M
,
BN
,
CP
respectively. Denote
K
K
K
as the circumcenter of the triangle formed by lines
A
b
A
c
,
B
c
B
a
,
C
a
C
b
.
A_bA_c,B_cB_a,C_aC_b.
A
b
A
c
,
B
c
B
a
,
C
a
C
b
.
Prove
O
H
/
/
I
K
OH//IK
O
H
//
I
K
.
2
1
Hide problems
Sequence number theory
Given are the integers
a
,
b
,
c
,
α
,
β
a , b , c, \alpha, \beta
a
,
b
,
c
,
α
,
β
and the sequence
(
u
n
)
(u_n)
(
u
n
)
is defined by
u
1
=
α
,
u
2
=
β
,
u
n
+
2
=
a
u
n
+
1
+
b
u
n
+
c
u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c
u
1
=
α
,
u
2
=
β
,
u
n
+
2
=
a
u
n
+
1
+
b
u
n
+
c
for all
n
≥
1
n \geq 1
n
≥
1
. a) Prove that if
a
=
3
,
b
=
−
2
,
c
=
−
1
a = 3 , b= -2 , c = -1
a
=
3
,
b
=
−
2
,
c
=
−
1
then there are infinitely many pairs of integers
(
α
;
β
)
(\alpha ; \beta)
(
α
;
β
)
so that
u
2023
=
2
2022
u_{2023}=2^{2022}
u
2023
=
2
2022
. b) Prove that there exists a positive integer
n
0
n_0
n
0
such that only one of the following two statements is true:i) There are infinitely many positive integers
m
m
m
, such that
u
n
0
u
n
0
+
1
…
u
n
0
+
m
u_{n_0}u_{n_0+1}\ldots u_{n_0+m}
u
n
0
u
n
0
+
1
…
u
n
0
+
m
is divisible by
7
2023
7^{2023}
7
2023
or
1
7
2023
17^{2023}
1
7
2023
ii) There are infinitely many positive integers
k
k
k
so that
u
n
0
u
n
0
+
1
…
u
n
0
+
k
−
1
u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1
u
n
0
u
n
0
+
1
…
u
n
0
+
k
−
1
is divisible by
2023
2023
2023
4
1
Hide problems
Vietnamese geo with way too many points
Given is a triangle
A
B
C
ABC
A
BC
and let
D
D
D
be the midpoint the major arc
B
A
C
BAC
B
A
C
of its circumcircle. Let
M
,
N
M , N
M
,
N
be the midpoints of
A
B
,
A
C
AB , AC
A
B
,
A
C
and
J
,
E
,
F
J , E , F
J
,
E
,
F
are the touchpoints of the incircle
(
I
)
(I)
(
I
)
of
△
A
B
C
\triangle ABC
△
A
BC
with
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. The line
M
N
MN
MN
intersects
J
E
,
J
F
JE , JF
J
E
,
J
F
at
K
,
H
K , H
K
,
H
respectively;
I
J
IJ
I
J
intersects the circle
(
B
I
C
)
(BIC)
(
B
I
C
)
at
G
G
G
and
D
G
DG
D
G
intersects
(
B
I
C
)
(BIC)
(
B
I
C
)
at
T
T
T
. a) Prove that
J
A
JA
J
A
passes through the midpoint of
H
K
HK
HK
and is perpendicular to
I
T
IT
I
T
. b) Let
R
,
S
R, S
R
,
S
respectively be the perpendicular projection of
D
D
D
on
A
B
,
A
C
AB, AC
A
B
,
A
C
. Take the points
P
,
Q
P, Q
P
,
Q
on
I
F
,
I
E
IF , IE
I
F
,
I
E
respectively such that
K
P
KP
K
P
and
H
Q
HQ
H
Q
are both perpendicular to
M
N
MN
MN
. Prove that the three lines
M
P
,
N
Q
MP , NQ
MP
,
NQ
and
R
S
RS
RS
are concurrent .
3
1
Hide problems
2023 VMO Problem 3
Find the maximum value of the positive real number
k
k
k
such that the inequality
1
k
a
b
+
c
2
+
1
k
b
c
+
a
2
+
1
k
c
a
+
b
2
≥
k
+
3
a
2
+
b
2
+
c
2
\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2}
kab
+
c
2
1
+
kb
c
+
a
2
1
+
k
c
a
+
b
2
1
≥
a
2
+
b
2
+
c
2
k
+
3
holds for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
2
+
b
2
+
c
2
=
2
(
a
b
+
b
c
+
c
a
)
.
a^2+b^2+c^2=2(ab+bc+ca).
a
2
+
b
2
+
c
2
=
2
(
ab
+
b
c
+
c
a
)
.