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Brazil Undergrad MO
2021 Brazil Undergrad MO
2021 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
Problem 3
1
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Almost Square-minus-constant
Find all positive integers
k
k
k
for which there is an irrational
α
>
1
\alpha>1
α
>
1
and a positive integer
N
N
N
such that
⌊
α
n
⌋
\left\lfloor\alpha^{n}\right\rfloor
⌊
α
n
⌋
is of the form
m
2
−
k
m^2-k
m
2
−
k
com
m
∈
Z
m \in \mathbb{Z}
m
∈
Z
for every integer
n
>
N
n>N
n
>
N
.
Problem 6
1
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Pairs whose concatenation form and almost-palindrome
We recursively define a set of goody pairs of words on the alphabet
{
a
,
b
}
\{a,b\}
{
a
,
b
}
as follows:-
(
a
,
b
)
(a,b)
(
a
,
b
)
is a goody pair; -
(
α
,
β
)
≠
(
a
,
b
)
(\alpha, \beta) \not= (a,b)
(
α
,
β
)
=
(
a
,
b
)
is a goody pair if and only if there is a goody pair
(
u
,
v
)
(u,v)
(
u
,
v
)
such that
(
α
,
β
)
=
(
u
v
,
v
)
(\alpha, \beta) = (uv,v)
(
α
,
β
)
=
(
uv
,
v
)
or
(
α
,
β
)
=
(
u
,
u
v
)
(\alpha, \beta) = (u,uv)
(
α
,
β
)
=
(
u
,
uv
)
Show that if
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
is a good pair then there exists a palindrome
γ
\gamma
γ
(possibly empty) such that
α
β
=
a
γ
b
\alpha\beta = a \gamma b
α
β
=
aγb
Problem 5
1
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What are the possible eigenvalues of a non-negative matrix?
Find all triplets
(
λ
1
,
λ
2
,
λ
3
)
∈
R
3
(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3
(
λ
1
,
λ
2
,
λ
3
)
∈
R
3
such that there exists a matrix
A
3
×
3
A_{3 \times 3}
A
3
×
3
with all entries being non-negative reals whose eigenvalues are
λ
1
,
λ
2
,
λ
3
\lambda_1,\lambda_2,\lambda_3
λ
1
,
λ
2
,
λ
3
.
Problem 4
1
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What is the mean of exponents of all naturals?
For every positive integeer
n
>
1
n>1
n
>
1
, let
k
(
n
)
k(n)
k
(
n
)
the largest positive integer
k
k
k
such that there exists a positive integer
m
m
m
such that
n
=
m
k
n = m^k
n
=
m
k
.Find
l
i
m
n
→
∞
∑
j
=
2
j
=
n
+
1
k
(
j
)
n
lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}
l
i
m
n
→
∞
n
∑
j
=
2
j
=
n
+
1
k
(
j
)
Problem 2
1
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f(x)^2=f(x sqrt(2))
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
from
C
2
C^2
C
2
(id est,
f
f
f
is twice differentiable and
f
′
′
f''
f
′′
is continuous.) such that for every real number
t
t
t
we have
f
(
t
)
2
=
f
(
t
2
)
f(t)^2=f(t \sqrt{2})
f
(
t
)
2
=
f
(
t
2
)
.
Problem 1
1
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Unitary matrices with rational entries
Consider the matrices like
M
=
(
a
b
c
c
a
b
b
c
a
)
M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)
M
=
a
c
b
b
a
c
c
b
a
such that
d
e
t
(
M
)
=
1
det(M) = 1
d
e
t
(
M
)
=
1
.Show thata) There are infinitely many matrices like above with
a
,
b
,
c
∈
Q
a,b,c \in \mathbb{Q}
a
,
b
,
c
∈
Q
b) There are finitely many matrices like above with
a
,
b
,
c
∈
Z
a,b,c \in \mathbb{Z}
a
,
b
,
c
∈
Z