MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2022 Vietnam National Olympiad
2022 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
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VMO 2022 problem 4 day 1
For every pair of positive integers
(
n
,
m
)
(n,m)
(
n
,
m
)
with
n
<
m
n<m
n
<
m
, denote
s
(
n
,
m
)
s(n,m)
s
(
n
,
m
)
be the number of positive integers such that the number is in the range
[
n
,
m
]
[n,m]
[
n
,
m
]
and the number is coprime with
m
m
m
. Find all positive integers
m
≥
2
m\ge 2
m
≥
2
such that
m
m
m
satisfy these condition: i)
s
(
n
,
m
)
m
−
n
≥
s
(
1
,
m
)
m
\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}
m
−
n
s
(
n
,
m
)
≥
m
s
(
1
,
m
)
for all
n
=
1
,
2
,
.
.
.
,
m
−
1
n=1,2,...,m-1
n
=
1
,
2
,
...
,
m
−
1
; ii)
202
2
m
+
1
2022^m+1
202
2
m
+
1
is divisible by
m
2
m^2
m
2
3
2
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VMO 2022 problem 3 day 1
Let
A
B
C
ABC
A
BC
be a triangle. Point
E
,
F
E,F
E
,
F
moves on the opposite ray of
B
A
,
C
A
BA,CA
B
A
,
C
A
such that
B
F
=
C
E
BF=CE
BF
=
CE
. Let
M
,
N
M,N
M
,
N
be the midpoint of
B
E
,
C
F
BE,CF
BE
,
CF
.
B
F
BF
BF
cuts
C
E
CE
CE
at
D
D
D
a) Suppost that
I
I
I
is the circumcenter of
(
D
B
E
)
(DBE)
(
D
BE
)
and
J
J
J
is the circumcenter of
(
D
C
F
)
(DCF)
(
D
CF
)
, Prove that
M
N
∥
I
J
MN \parallel IJ
MN
∥
I
J
b) Let
K
K
K
be the midpoint of
M
N
MN
MN
and
H
H
H
be the orthocenter of triangle
A
E
F
AEF
A
EF
. Prove that when
E
E
E
varies on the opposite ray of
B
A
BA
B
A
,
H
K
HK
HK
go through a fixed point
VMO 2022 problem 7 day 2
Let
A
B
C
ABC
A
BC
be an acute triangle,
B
,
C
B,C
B
,
C
fixed,
A
A
A
moves on the big arc
B
C
BC
BC
of
(
A
B
C
)
(ABC)
(
A
BC
)
. Let
O
O
O
be the circumcenter of
(
A
B
C
)
(ABC)
(
A
BC
)
(
B
,
O
,
C
(B,O,C
(
B
,
O
,
C
are not collinear,
A
B
≠
A
C
)
AB \ne AC)
A
B
=
A
C
)
,
(
I
)
(I)
(
I
)
is the incircle of triangle
A
B
C
ABC
A
BC
.
(
I
)
(I)
(
I
)
tangents to
B
C
BC
BC
at
D
D
D
. Let
I
a
I_a
I
a
be the
A
A
A
-excenter of triangle
A
B
C
ABC
A
BC
.
I
a
D
I_aD
I
a
D
cuts
O
I
OI
O
I
at
L
L
L
. Let
E
E
E
lies on
(
I
)
(I)
(
I
)
such that
D
E
∥
A
I
DE \parallel AI
D
E
∥
A
I
. a)
L
E
LE
L
E
cuts
A
I
AI
A
I
at
F
F
F
. Prove that
A
F
=
A
I
AF=AI
A
F
=
A
I
. b) Let
M
M
M
lies on the circle
(
J
)
(J)
(
J
)
go through
I
a
,
B
,
C
I_a,B,C
I
a
,
B
,
C
such that
I
a
M
∥
A
D
I_aM \parallel AD
I
a
M
∥
A
D
.
M
D
MD
M
D
cuts
(
J
)
(J)
(
J
)
again at
N
N
N
. Prove that the midpoint
T
T
T
of
M
N
MN
MN
lies on a fixed circle.
2
2
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VMO 2022 problem 2 day 1
Find all function
f
:
R
+
→
R
+
f:\mathbb R^+ \rightarrow \mathbb R^+
f
:
R
+
→
R
+
such that: f\left(\frac{f(x)}{x}+y\right)=1+f(y), \forall x,y \in \mathbb R^+.
VMO 2022 problem 6 day 2
We are given 4 similar dices. Denote
x
i
(
1
≤
x
i
≤
6
)
x_i (1\le x_i \le 6)
x
i
(
1
≤
x
i
≤
6
)
be the number of dots on a face appearing on the
i
i
i
-th dice
1
≤
i
≤
4
1\le i \le 4
1
≤
i
≤
4
a) Find the numbers of
(
x
1
,
x
2
,
x
3
,
x
4
)
(x_1,x_2,x_3,x_4)
(
x
1
,
x
2
,
x
3
,
x
4
)
b) Find the probability that there is a number
x
j
x_j
x
j
such that
x
j
x_j
x
j
is equal to the sum of the other 3 numbers c) Find the probability that we can divide
x
1
,
x
2
,
x
3
,
x
4
x_1,x_2,x_3,x_4
x
1
,
x
2
,
x
3
,
x
4
into 2 groups has the same sum
1
2
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VMO 2022 problem 1 day 1
Let
a
a
a
be a non-negative real number and a sequence
(
u
n
)
(u_n)
(
u
n
)
defined as:
u
1
=
6
,
u
n
+
1
=
2
n
+
a
n
+
n
+
a
n
u
n
+
4
,
∀
n
≥
1
u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1
u
1
=
6
,
u
n
+
1
=
n
2
n
+
a
+
n
n
+
a
u
n
+
4
,
∀
n
≥
1
a) With
a
=
0
a=0
a
=
0
, prove that there exist a finite limit of
(
u
n
)
(u_n)
(
u
n
)
and find that limit b) With
a
≥
0
a \ge 0
a
≥
0
, prove that there exist a finite limit of
(
u
n
)
(u_n)
(
u
n
)
VMO 2022 problem 5 day 2
Consider 2 non-constant polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
, with nonnegative coefficients. The coefficients of
P
(
x
)
P(x)
P
(
x
)
is not larger than
2021
2021
2021
and
Q
(
x
)
Q(x)
Q
(
x
)
has at least one coefficient larger than
2021
2021
2021
. Assume that
P
(
2022
)
=
Q
(
2022
)
P(2022)=Q(2022)
P
(
2022
)
=
Q
(
2022
)
and
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
has a root
p
q
≠
0
(
p
,
q
∈
Z
,
(
p
,
q
)
=
1
)
\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)
q
p
=
0
(
p
,
q
∈
Z
,
(
p
,
q
)
=
1
)
. Prove that
∣
p
∣
+
n
∣
q
∣
≤
Q
(
n
)
−
P
(
n
)
|p|+n|q|\le Q(n)-P(n)
∣
p
∣
+
n
∣
q
∣
≤
Q
(
n
)
−
P
(
n
)
for all
n
=
1
,
2
,
.
.
.
,
2021
n=1,2,...,2021
n
=
1
,
2
,
...
,
2021