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Undergraduate contests
Brazil Undergrad MO
2022 Brazil Undergrad MO
2022 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
3
1
Hide problems
a sequence that becomes self-referencing
Let
(
a
n
)
n
∈
N
(a_n)_{n \in \mathbb{N}}
(
a
n
)
n
∈
N
be a sequence of integers. Define
a
n
(
0
)
=
a
n
a_n^{(0)} = a_n
a
n
(
0
)
=
a
n
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
. For all
M
≥
0
M \geq 0
M
≥
0
, we define
(
a
n
(
M
+
1
)
)
n
∈
N
:
a
n
(
M
+
1
)
=
a
n
+
1
(
M
)
−
a
n
(
M
)
,
∀
n
∈
N
(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}
(
a
n
(
M
+
1
)
)
n
∈
N
:
a
n
(
M
+
1
)
=
a
n
+
1
(
M
)
−
a
n
(
M
)
,
∀
n
∈
N
. We say that
(
a
n
)
n
∈
N
(a_n)_{n \in \mathbb{N}}
(
a
n
)
n
∈
N
is
(M + 1)-self-referencing
\textrm{(M + 1)-self-referencing}
(M + 1)-self-referencing
if there exists
k
1
k_1
k
1
and
k
2
k_2
k
2
fixed positive integers such that
a
n
+
k
1
=
a
n
+
k
2
(
M
+
1
)
,
∀
n
∈
N
a_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}
a
n
+
k
1
=
a
n
+
k
2
(
M
+
1
)
,
∀
n
∈
N
.(a) Does there exist a sequence of integers such that the smallest
M
M
M
such that it is
M-self-referencing
\textrm{M-self-referencing}
M-self-referencing
is
M
=
2022
M = 2022
M
=
2022
?(a) Does there exist a stricly positive sequence of integers such that the smallest
M
M
M
such that it is
M-self-referencing
\textrm{M-self-referencing}
M-self-referencing
is
M
=
2022
M = 2022
M
=
2022
?
5
1
Hide problems
Product of density-like function is larger than 1/4
Given
X
⊂
N
X \subset \mathbb{N}
X
⊂
N
, define
d
(
X
)
d(X)
d
(
X
)
as the largest
c
∈
[
0
,
1
]
c \in [0, 1]
c
∈
[
0
,
1
]
such that for any
a
<
c
a < c
a
<
c
and
n
0
∈
N
n_0\in \mathbb{N}
n
0
∈
N
, there exists
m
,
r
∈
N
m, r \in \mathbb{N}
m
,
r
∈
N
with
r
≥
n
0
r \geq n_0
r
≥
n
0
and
∣
X
∩
[
m
,
m
+
r
)
∣
r
≥
a
\frac{\mid X \cap [m, m+r)\mid}{r} \geq a
r
∣
X
∩
[
m
,
m
+
r
)
∣
≥
a
. Let
E
,
F
⊂
N
E, F \subset \mathbb{N}
E
,
F
⊂
N
such that
d
(
E
)
d
(
F
)
>
1
/
4
d(E)d(F) > 1/4
d
(
E
)
d
(
F
)
>
1/4
. Prove that for any prime
p
p
p
and
k
∈
N
k\in\mathbb{N}
k
∈
N
, there exists
m
∈
E
,
n
∈
F
m \in E, n \in F
m
∈
E
,
n
∈
F
such that
m
≡
n
(
m
o
d
p
k
)
m\equiv n \pmod{p^k}
m
≡
n
(
mod
p
k
)
6
1
Hide problems
Showing tan((p+1) theta) has numerator divisible by p
Let
p
≡
3
(
mod
4
)
p \equiv 3 \,(\textrm{mod}\, 4)
p
≡
3
(
mod
4
)
be a prime and
θ
\theta
θ
some angle such that
tan
(
θ
)
\tan(\theta)
tan
(
θ
)
is rational. Prove that
tan
(
(
p
+
1
)
θ
)
\tan((p+1)\theta)
tan
((
p
+
1
)
θ
)
is a rational number with numerator divisible by
p
p
p
, that is,
tan
(
(
p
+
1
)
θ
)
=
u
v
\tan((p+1)\theta) = \frac{u}{v}
tan
((
p
+
1
)
θ
)
=
v
u
with
u
,
v
∈
Z
,
v
>
0
,
mdc
(
u
,
v
)
=
1
u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1
u
,
v
∈
Z
,
v
>
0
,
mdc
(
u
,
v
)
=
1
and
u
≡
0
(
mod
p
)
u \equiv 0 \,(\textrm{mod}\,p)
u
≡
0
(
mod
p
)
.
2
1
Hide problems
Two matrices that generate a whole family
Let
G
G
G
be the set of
2
×
2
2\times 2
2
×
2
matrices that such
G
=
{
(
a
b
c
d
)
∣
a
,
b
,
c
,
d
∈
Z
,
a
d
−
b
c
=
1
,
c
is a multiple of
3
}
G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid\, a,b,c,d \in \mathbb{Z}, ad-bc = 1, c \text{ is a multiple of } 3 \right\}
G
=
{
(
a
c
b
d
)
∣
a
,
b
,
c
,
d
∈
Z
,
a
d
−
b
c
=
1
,
c
is a multiple of
3
}
and two matrices in
G
G
G
:
A
=
(
1
1
0
1
)
B
=
(
−
1
1
−
3
2
)
A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\;\;\; B = \begin{pmatrix} -1 & 1 \\ -3 & 2 \end{pmatrix}
A
=
(
1
0
1
1
)
B
=
(
−
1
−
3
1
2
)
Show that any matrix in
G
G
G
can be written as a product
M
1
M
2
⋯
M
r
M_1M_2\cdots M_r
M
1
M
2
⋯
M
r
such that
M
i
∈
{
A
,
A
−
1
,
B
,
B
−
1
}
,
∀
i
≤
r
M_i \in \{A, A^{-1}, B, B^{-1}\}, \forall i \leq r
M
i
∈
{
A
,
A
−
1
,
B
,
B
−
1
}
,
∀
i
≤
r
1
1
Hide problems
Functional analysis
Let
0
<
a
<
1
0<a<1
0
<
a
<
1
. Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
continuous at
x
=
0
x = 0
x
=
0
such that
f
(
x
)
+
f
(
a
x
)
=
x
,
∀
x
∈
R
f(x) + f(ax) = x,\, \forall x \in \mathbb{R}
f
(
x
)
+
f
(
a
x
)
=
x
,
∀
x
∈
R
4
1
Hide problems
Convergence of series Brazilian MO 2022 P4
Let
α
,
c
>
0
\alpha, c > 0
α
,
c
>
0
, define
x
1
=
c
x_1 = c
x
1
=
c
and let
x
n
+
1
=
x
n
e
−
x
n
α
x_{n + 1} = x_n e^{-x_n^\alpha}
x
n
+
1
=
x
n
e
−
x
n
α
for
n
≥
1
n \geq 1
n
≥
1
. For which values of
β
\beta
β
does
∑
i
=
1
∞
x
n
β
\sum_{i = 1}^{\infty} x_n^\beta
∑
i
=
1
∞
x
n
β
converge?