MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2024 Vietnam National Olympiad
2024 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(7)
7
1
Hide problems
Assigning numbers on the edges of a polyhedron
In the space, there is a convex polyhedron
D
D
D
such that for every vertex of
D
D
D
, there are an even number of edges passing through that vertex. We choose a face
F
F
F
of
D
D
D
. Then we assign each edge of
D
D
D
a positive integer such that for all faces of
D
D
D
different from
F
F
F
, the sum of the numbers assigned on the edges of that face is a positive integer divisible by
2024
2024
2024
. Prove that the sum of the numbers assigned on the edges of
F
F
F
is also a positive integer divisible by
2024
2024
2024
.
6
1
Hide problems
We're back with arithmetic functions!
For each positive integer
n
n
n
, let
τ
(
n
)
\tau (n)
τ
(
n
)
be the number of positive divisors of
n
n
n
.a) Find all positive integers
n
n
n
such that
τ
(
n
)
+
2023
=
n
\tau(n)+2023=n
τ
(
n
)
+
2023
=
n
. b) Prove that there exist infinitely many positive integers
k
k
k
such that there are exactly two positive integers
n
n
n
satisfying
τ
(
k
n
)
+
2023
=
n
\tau(kn)+2023=n
τ
(
kn
)
+
2023
=
n
.
5
1
Hide problems
Polynomials having a large number of roots!
For each polynomial
P
(
x
)
P(x)
P
(
x
)
, define
P
1
(
x
)
=
P
(
x
)
,
∀
x
∈
R
,
P_1(x)=P(x), \forall x \in \mathbb{R},
P
1
(
x
)
=
P
(
x
)
,
∀
x
∈
R
,
P
2
(
x
)
=
P
(
P
1
(
x
)
)
,
∀
x
∈
R
,
P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},
P
2
(
x
)
=
P
(
P
1
(
x
))
,
∀
x
∈
R
,
.
.
.
...
...
P
2024
(
x
)
=
P
(
P
2023
(
x
)
)
,
∀
x
∈
R
.
P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.
P
2024
(
x
)
=
P
(
P
2023
(
x
))
,
∀
x
∈
R
.
Let
a
>
2
a>2
a
>
2
be a real number. Is there a polynomial
P
P
P
with real coefficients such that for all
t
∈
(
−
a
,
a
)
t \in (-a, a)
t
∈
(
−
a
,
a
)
, the equation
P
2024
(
x
)
=
t
P_{2024}(x)=t
P
2024
(
x
)
=
t
has
2
2024
2^{2024}
2
2024
distinct real roots?
4
1
Hide problems
Moving marbles...
k
k
k
marbles are placed onto the cells of a
2024
×
2024
2024 \times 2024
2024
×
2024
grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).a) Assume that
k
=
2024
k=2024
k
=
2024
. Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions. b) Determine the largest value of
k
k
k
such that for all arrangements of
k
k
k
marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.
3
1
Hide problems
Geometry but... lots of points!
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcenter
O
O
O
. Let
A
′
A'
A
′
be the center of the circle passing through
C
C
C
and tangent to
A
B
AB
A
B
at
A
A
A
, let
B
′
B'
B
′
be the center of the circle passing through
A
A
A
and tangent to
B
C
BC
BC
at
B
B
B
, let
C
′
C'
C
′
be the center of the circle passing through
B
B
B
and tangent to
C
A
CA
C
A
at
C
C
C
.a) Prove that the area of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is not less than the area of triangle
A
B
C
ABC
A
BC
. b) Let
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
be the projections of
O
O
O
onto lines
A
′
B
′
,
B
′
C
′
,
C
′
A
′
A'B', B'C', C'A'
A
′
B
′
,
B
′
C
′
,
C
′
A
′
. Given that the circumcircle of triangle
X
Y
Z
XYZ
X
Y
Z
intersects lines
A
′
B
′
,
B
′
C
′
,
C
′
A
′
A'B', B'C', C'A'
A
′
B
′
,
B
′
C
′
,
C
′
A
′
again at
X
′
,
Y
′
,
Z
′
X', Y', Z'
X
′
,
Y
′
,
Z
′
(
X
′
≠
X
,
Y
′
≠
Y
,
Z
′
≠
Z
X' \neq X, Y' \neq Y, Z' \neq Z
X
′
=
X
,
Y
′
=
Y
,
Z
′
=
Z
), prove that lines
A
X
′
,
B
Y
′
,
C
Z
′
AX', BY', CZ'
A
X
′
,
B
Y
′
,
C
Z
′
are concurrent.
2
1
Hide problems
Polynomial is back at VMO!
Find all polynomials
P
(
x
)
,
Q
(
x
)
P(x), Q(x)
P
(
x
)
,
Q
(
x
)
with real coefficients such that for all real numbers
a
a
a
,
P
(
a
)
P(a)
P
(
a
)
is a root of the equation
x
2023
+
Q
(
a
)
x
2
+
(
a
2024
+
a
)
x
+
a
3
+
2025
a
=
0.
x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.
x
2023
+
Q
(
a
)
x
2
+
(
a
2024
+
a
)
x
+
a
3
+
2025
a
=
0.
1
1
Hide problems
Sequences, polynomials and limits...
For each real number
x
x
x
, let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denote the largest integer not exceeding
x
x
x
. A sequence
{
a
n
}
n
=
1
∞
\{a_n \}_{n=1}^{\infty}
{
a
n
}
n
=
1
∞
is defined by
a
n
=
1
4
⌊
−
log
4
n
⌋
,
∀
n
≥
1.
a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.
a
n
=
4
⌊
−
l
o
g
4
n
⌋
1
,
∀
n
≥
1.
Let
b
n
=
1
n
2
(
∑
k
=
1
n
a
k
−
1
a
1
+
a
2
)
,
∀
n
≥
1.
b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.
b
n
=
n
2
1
(
∑
k
=
1
n
a
k
−
a
1
+
a
2
1
)
,
∀
n
≥
1.
a) Find a polynomial
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
b
n
=
P
(
a
n
n
)
,
∀
n
≥
1
b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1
b
n
=
P
(
n
a
n
)
,
∀
n
≥
1
. b) Prove that there exists a strictly increasing sequence
{
n
k
}
k
=
1
∞
\{n_k \}_{k=1}^{\infty}
{
n
k
}
k
=
1
∞
of positive integers such that
lim
k
→
∞
b
n
k
=
2024
2025
.
\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.
k
→
∞
lim
b
n
k
=
2025
2024
.