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Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2022 Vietnam TST
2022 Vietnam TST
Part of
Vietnam Team Selection Test
Subcontests
(6)
6
1
Hide problems
Vietnam TST #6
Given a set
A
=
{
1
;
2
;
.
.
.
;
4044
}
A=\{1;2;...;4044\}
A
=
{
1
;
2
;
...
;
4044
}
. They color
2022
2022
2022
numbers of them by white and the rest of them by black. With each
i
∈
A
i\in A
i
∈
A
, called the important number of
i
i
i
be the number of all white numbers smaller than
i
i
i
and black numbers larger than
i
i
i
. With every natural number
m
m
m
, find all positive integers
k
k
k
that exist a way to color the numbers that can get
k
k
k
important numbers equal to
m
m
m
.
5
1
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Vietnam TST #5
A fractional number
x
x
x
is called pretty if it has finite expression in base
−
b
-b
−
b
numeral system,
b
b
b
is a positive integer in
[
2
;
2022
]
[2;2022]
[
2
;
2022
]
. Prove that there exists finite positive integers
n
≥
4
n\geq 4
n
≥
4
that with every
m
m
m
in
(
2
n
3
;
n
)
(\frac{2n}{3}; n)
(
3
2
n
;
n
)
then there is at least one pretty number between
m
n
−
m
\frac{m}{n-m}
n
−
m
m
and
n
−
m
m
\frac{n-m}{m}
m
n
−
m
4
1
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Vietnam TST #4
An acute, non-isosceles triangle
A
B
C
ABC
A
BC
is inscribed in a circle with centre
O
O
O
. A line go through
O
O
O
and midpoint
I
I
I
of
B
C
BC
BC
intersects
A
B
,
A
C
AB, AC
A
B
,
A
C
at
E
,
F
E, F
E
,
F
respectively. Let
D
,
G
D, G
D
,
G
be reflections to
A
A
A
over
O
O
O
and circumcentre of
(
A
E
F
)
(AEF)
(
A
EF
)
, respectively. Let
K
K
K
be the reflection of
O
O
O
over circumcentre of
(
O
B
C
)
(OBC)
(
OBC
)
.
a
)
a)
a
)
Prove that
D
,
G
,
K
D, G, K
D
,
G
,
K
are collinear.
b
)
b)
b
)
Let
M
,
N
M, N
M
,
N
are points on
K
B
,
K
C
KB, KC
K
B
,
K
C
that
I
M
⊥
A
C
IM\perp AC
I
M
⊥
A
C
,
I
N
⊥
A
B
IN\perp AB
I
N
⊥
A
B
. The midperpendiculars of
I
K
IK
I
K
intersects
M
N
MN
MN
at
H
H
H
. Assume that
I
H
IH
I
H
intersects
A
B
,
A
C
AB, AC
A
B
,
A
C
at
P
,
Q
P, Q
P
,
Q
respectively. Prove that the circumcircle of
△
A
P
Q
\triangle APQ
△
A
PQ
intersects
(
O
)
(O)
(
O
)
the second time at a point on
A
I
AI
A
I
.
3
1
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Vietnam TST #3
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram,
A
C
AC
A
C
intersects
B
D
BD
B
D
at
I
I
I
. Consider point
G
G
G
inside
△
A
B
C
\triangle ABC
△
A
BC
that satisfy
∠
I
A
G
=
∠
I
B
G
≠
4
5
∘
−
∠
A
I
B
4
\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}
∠
I
A
G
=
∠
I
BG
=
4
5
∘
−
4
∠
A
I
B
. Let
E
,
G
E,G
E
,
G
be projections of
C
C
C
on
A
G
AG
A
G
and
D
D
D
on
B
G
BG
BG
. The
E
−
E-
E
−
median line of
△
B
E
F
\triangle BEF
△
BEF
and
F
−
F-
F
−
median line of
△
A
E
F
\triangle AEF
△
A
EF
intersects at
H
H
H
.
a
)
a)
a
)
Prove that
A
F
,
B
E
AF,BE
A
F
,
BE
and
I
H
IH
I
H
concurrent. Call the concurrent point
L
L
L
.
b
)
b)
b
)
Let
K
K
K
be the intersection of
C
E
CE
CE
and
D
F
DF
D
F
. Let
J
J
J
circumcenter of
(
L
A
B
)
(LAB)
(
L
A
B
)
and
M
,
N
M,N
M
,
N
are respectively be circumcenters of
(
E
I
J
)
(EIJ)
(
E
I
J
)
and
(
F
I
J
)
(FIJ)
(
F
I
J
)
. Prove that
E
M
,
F
N
EM,FN
EM
,
FN
and the line go through circumcenters of
(
G
A
B
)
,
(
K
C
D
)
(GAB),(KCD)
(
G
A
B
)
,
(
K
C
D
)
are concurrent.
2
1
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Vietnam TST #2
Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number
26
;
4
26; 4
26
;
4
and
2022
2022
2022
(each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.
1
1
Hide problems
Vietnam TST #1
Given a real number
α
\alpha
α
and consider function
φ
(
x
)
=
x
2
e
α
x
\varphi(x)=x^2e^{\alpha x}
φ
(
x
)
=
x
2
e
αx
for
x
∈
R
x\in\mathbb R
x
∈
R
. Find all function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
that satisfy:
f
(
φ
(
x
)
+
f
(
y
)
)
=
y
+
φ
(
f
(
x
)
)
f(\varphi(x)+f(y))=y+\varphi(f(x))
f
(
φ
(
x
)
+
f
(
y
))
=
y
+
φ
(
f
(
x
))
forall
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R