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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2024 Vietnam Team Selection Test
2024 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
5
1
Hide problems
Equal angles
Let incircle
(
I
)
(I)
(
I
)
of triangle
A
B
C
ABC
A
BC
touch the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
(
O
)
(O)
(
O
)
be the circumcircle of
A
B
C
ABC
A
BC
. Ray
E
F
EF
EF
meets
(
O
)
(O)
(
O
)
at
M
M
M
. Tangents at
M
M
M
and
A
A
A
of
(
O
)
(O)
(
O
)
meet at
S
S
S
. Tangents at
B
B
B
and
C
C
C
of
(
O
)
(O)
(
O
)
meet at
T
T
T
. Line
T
I
TI
T
I
meets
O
A
OA
O
A
at
J
J
J
. Prove that
∠
A
S
J
=
∠
I
S
T
\angle ASJ=\angle IST
∠
A
S
J
=
∠
I
ST
.
6
1
Hide problems
Old idea in a new context
Let
P
(
x
)
∈
Z
[
x
]
P(x) \in \mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
be a polynomial. Determine all polynomials
Q
(
x
)
∈
Z
[
x
]
Q(x) \in \mathbb{Z}[x]
Q
(
x
)
∈
Z
[
x
]
, such that for every positive integer
n
n
n
, there exists a polynomial
R
n
(
x
)
∈
Z
[
x
]
R_n(x) \in \mathbb{Z}[x]
R
n
(
x
)
∈
Z
[
x
]
satisfies
Q
(
x
)
2
n
−
1
=
R
n
(
x
)
(
P
(
x
)
2
n
−
1
)
.
Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).
Q
(
x
)
2
n
−
1
=
R
n
(
x
)
(
P
(
x
)
2
n
−
1
)
.
4
1
Hide problems
A weird polynomial inequality
Let
α
∈
(
1
,
+
∞
)
\alpha \in (1, +\infty)
α
∈
(
1
,
+
∞
)
be a real number, and let
P
(
x
)
∈
R
[
x
]
P(x) \in \mathbb{R}[x]
P
(
x
)
∈
R
[
x
]
be a monic polynomial with degree
24
24
24
, such that (i)
P
(
0
)
=
1
P(0) = 1
P
(
0
)
=
1
. (ii)
P
(
x
)
P(x)
P
(
x
)
has exactly
24
24
24
positive real roots that are all less than or equal to
α
\alpha
α
. Show that
∣
P
(
1
)
∣
≤
(
19
5
)
5
(
α
−
1
)
24
|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}
∣
P
(
1
)
∣
≤
(
5
19
)
5
(
α
−
1
)
24
.
2
1
Hide problems
Spring flower garden
In a garden, which is organized as a
2024
×
2024
2024\times 2024
2024
×
2024
board, we plant three types of flowers: roses, daisies, and orchids. We want to plant flowers such that the following conditions are satisfied:(i) Each grid is planted with at most one type of flower. Some grids can be left blank and not planted. (ii) For each planted grid
A
A
A
, there exist exactly
3
3
3
other planted grids in the same column or row such that those
3
3
3
grids are planted with flowers of different types from
A
A
A
's. (iii) Each flower is planted in at least
1
1
1
grid. What is the maximal number of the grids that can be planted with flowers?
1
1
Hide problems
Functional equation with polynomial component
Let
P
(
x
)
∈
R
[
x
]
P(x) \in \mathbb{R}[x]
P
(
x
)
∈
R
[
x
]
be a monic, non-constant polynomial. Determine all continuous functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
P
(
x
)
)
+
y
+
2023
f
(
y
)
)
=
P
(
x
)
+
2024
f
(
y
)
,
f(f(P(x))+y+2023f(y))=P(x)+2024f(y),
f
(
f
(
P
(
x
))
+
y
+
2023
f
(
y
))
=
P
(
x
)
+
2024
f
(
y
)
,
for all reals
x
,
y
x,y
x
,
y
.
3
1
Hide problems
Similar triangles
Let
A
B
C
ABC
A
BC
be an acute scalene triangle. Incircle of
A
B
C
ABC
A
BC
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
be feet the altitudes of from
A
,
B
,
C
A,B,C
A
,
B
,
C
to the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the reflections of
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
in
E
F
,
F
D
,
D
E
EF,FD,DE
EF
,
F
D
,
D
E
respectively. Prove that triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
are similar.