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Brazil Undergrad MO
2023 Brazil Undergrad MO
2023 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
6
1
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For which constants does this sequence exist?
Determine all pairs
(
c
,
d
)
∈
R
2
(c, d) \in \mathbb{R}^2
(
c
,
d
)
∈
R
2
of real constants such that there is a sequence
(
a
n
)
n
≥
1
(a_n)_{n\geq1}
(
a
n
)
n
≥
1
of positive real numbers such that, for all
n
≥
1
n \geq 1
n
≥
1
,
a
n
≥
c
⋅
a
n
+
1
+
d
⋅
∑
1
≤
j
<
n
a
j
.
a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .
a
n
≥
c
⋅
a
n
+
1
+
d
⋅
1
≤
j
<
n
∑
a
j
.
5
1
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Easy problem, but with infinite calculations
A drunken horse moves on an infinite board whose squares are numbered in pairs
(
a
,
b
)
∈
Z
2
(a, b) \in \mathbb{Z}^2
(
a
,
b
)
∈
Z
2
. In each movement, the 8 possibilities
(
a
,
b
)
→
(
a
±
1
,
b
±
2
)
,
(a, b) \rightarrow (a \pm 1, b \pm 2),
(
a
,
b
)
→
(
a
±
1
,
b
±
2
)
,
(
a
,
b
)
→
(
a
±
2
,
b
±
1
)
(a, b) \rightarrow (a \pm 2, b \pm 1)
(
a
,
b
)
→
(
a
±
2
,
b
±
1
)
are equally likely. Knowing that the knight starts at
(
0
,
0
)
(0, 0)
(
0
,
0
)
, calculate the probability that, after
2023
2023
2023
moves, it is in a square
(
a
,
b
)
(a, b)
(
a
,
b
)
with
a
≡
4
(
m
o
d
8
)
a \equiv 4 \pmod 8
a
≡
4
(
mod
8
)
and
b
≡
5
(
m
o
d
8
)
b \equiv 5 \pmod 8
b
≡
5
(
mod
8
)
.
4
1
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Integer matrices known fact
Let
M
2
(
Z
)
M_2(\mathbb{Z})
M
2
(
Z
)
be the set of
2
×
2
2 \times 2
2
×
2
matrices with integer entries. Let
A
∈
M
2
(
Z
)
A \in M_2(\mathbb{Z})
A
∈
M
2
(
Z
)
such that
A
2
+
5
I
=
0
,
A^2+5I=0,
A
2
+
5
I
=
0
,
where
I
∈
M
2
(
Z
)
I \in M_2(\mathbb{Z})
I
∈
M
2
(
Z
)
and
0
∈
M
2
(
Z
)
0 \in M_2(\mathbb{Z})
0
∈
M
2
(
Z
)
denote the identity and null matrices, respectively. Prove that there exists an invertible matrix
C
∈
M
2
(
Z
)
C \in M_2(\mathbb{Z})
C
∈
M
2
(
Z
)
with
C
−
1
∈
M
2
(
Z
)
C^{-1} \in M_2(\mathbb{Z})
C
−
1
∈
M
2
(
Z
)
such that
C
A
C
−
1
=
(
1
2
−
3
−
1
)
ou
C
A
C
−
1
=
(
0
1
−
5
0
)
.
CAC^{-1} = \begin{pmatrix} 1 & 2\\ -3 & -1 \end{pmatrix} \text{ ou } CAC^{-1} = \begin{pmatrix} 0 & 1\\ -5 & 0 \end{pmatrix}.
C
A
C
−
1
=
(
1
−
3
2
−
1
)
ou
C
A
C
−
1
=
(
0
−
5
1
0
)
.
3
1
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More analysis... Find a bound for sum of sqrt[k]{x}
Prove that there exists a constant
C
>
0
C > 0
C
>
0
such that, for any integers
m
,
n
m, n
m
,
n
with
n
≥
m
>
1
n \geq m > 1
n
≥
m
>
1
and any real number
x
>
1
x > 1
x
>
1
,
∑
k
=
m
n
x
k
≤
C
(
m
2
⋅
x
m
−
1
log
x
+
n
)
\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)
k
=
m
∑
n
k
x
≤
C
(
lo
g
x
m
2
⋅
m
−
1
x
+
n
)
2
1
Hide problems
find the sum of the inverses of the central binomial coefficients
Let
a
n
=
1
(
2
n
n
)
,
∀
n
≤
1
a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1
a
n
=
(
n
2
n
)
1
,
∀
n
≤
1
.a) Show that
∑
n
=
1
+
∞
a
n
x
n
\sum\limits_{n=1}^{+\infty}a_nx^n
n
=
1
∑
+
∞
a
n
x
n
converges for all
x
∈
(
−
4
,
4
)
x \in (-4, 4)
x
∈
(
−
4
,
4
)
and that the function
f
(
x
)
=
∑
n
=
1
+
∞
a
n
x
n
f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n
f
(
x
)
=
n
=
1
∑
+
∞
a
n
x
n
satisfies the differential equation
x
(
x
−
4
)
f
′
(
x
)
+
(
x
+
2
)
f
(
x
)
=
−
x
x(x - 4)f'(x) + (x + 2)f(x) = -x
x
(
x
−
4
)
f
′
(
x
)
+
(
x
+
2
)
f
(
x
)
=
−
x
.b) Prove that
∑
n
=
1
+
∞
1
(
2
n
n
)
=
1
3
+
2
π
3
27
\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}
n
=
1
∑
+
∞
(
n
2
n
)
1
=
3
1
+
27
2
π
3
.
1
1
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Is function p exponential?
Let
p
p
p
be the potentioral function, from positive integers to positive integers, defined by
p
(
1
)
=
1
p(1) = 1
p
(
1
)
=
1
and
p
(
n
+
1
)
=
p
(
n
)
p(n + 1) = p(n)
p
(
n
+
1
)
=
p
(
n
)
, if
n
+
1
n + 1
n
+
1
is not a perfect power and
p
(
n
+
1
)
=
(
n
+
1
)
⋅
p
(
n
)
p(n + 1) = (n + 1) \cdot p(n)
p
(
n
+
1
)
=
(
n
+
1
)
⋅
p
(
n
)
, otherwise. Is there a positive integer
N
N
N
such that, for all
n
>
N
,
n > N,
n
>
N
,
p
(
n
)
>
2
n
p(n) > 2^n
p
(
n
)
>
2
n
?