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Undergraduate contests
Brazil Undergrad MO
2020 Brazil Undergrad MO
2020 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
Problem 2
1
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(Not) Almost Fibonacci
For a positive integer
a
a
a
, define
F
1
(
a
)
=
1
F_1 ^{(a)}=1
F
1
(
a
)
=
1
,
F
2
(
a
)
=
a
F_2 ^{(a)}=a
F
2
(
a
)
=
a
and for
n
>
2
n>2
n
>
2
,
F
n
(
a
)
=
F
n
−
1
(
a
)
+
F
n
−
2
(
a
)
F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}
F
n
(
a
)
=
F
n
−
1
(
a
)
+
F
n
−
2
(
a
)
. A positive integer is fibonatic when it is equal to
F
n
(
a
)
F_n ^{(a)}
F
n
(
a
)
for a positive integer
a
a
a
and
n
>
3
n>3
n
>
3
. Prove that there are infintely many not fibonatic integers.
Problem 5
1
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Yet another combinatorial game (in a spaceship!)
Let
N
N
N
a positive integer.In a spaceship there are
2
⋅
N
2 \cdot N
2
⋅
N
people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows:1) The first alien chooses any person as she wishes.2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn.3) The alien that can't play in her turn loses.Prove that second player has a winning strategy if, and only if, the
2
⋅
N
2 \cdot N
2
⋅
N
people can be divided in
N
N
N
pairs in such a way that two people in the same pair are friends.
Problem 4
1
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Matrices as a sum of two squares matrices
For each of the following, provide proof or a counterexample:a) Every
2
×
2
2\times2
2
×
2
matrix with real entries can we written as the sum of the squares of two
2
×
2
2\times2
2
×
2
matrices with real entries. b) Every
3
×
3
3\times3
3
×
3
matrix with real entries can we written as the sum of the squares of two
3
×
3
3\times3
3
×
3
matrices with real entries.
Problem 6
1
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iterate function
Let
f
(
x
)
=
2
x
2
+
x
−
1
,
f
0
(
x
)
=
x
f(x) = 2x^2 + x - 1, f^{0}(x) = x
f
(
x
)
=
2
x
2
+
x
−
1
,
f
0
(
x
)
=
x
, and
f
n
+
1
(
x
)
=
f
(
f
n
(
x
)
)
f^{n+1}(x) = f(f^{n}(x))
f
n
+
1
(
x
)
=
f
(
f
n
(
x
))
for all real
x
>
0
x>0
x
>
0
and
n
≥
0
n \ge 0
n
≥
0
integer (that is,
f
n
f^{n}
f
n
is
f
f
f
iterated
n
n
n
times).a) Find the number of distinct real roots of the equation
f
3
(
x
)
=
x
f^{3}(x) = x
f
3
(
x
)
=
x
b) Find, for each
n
≥
0
n \ge 0
n
≥
0
integer, the number of distinct real solutions of the equation
f
n
(
x
)
=
0
f^{n}(x) = 0
f
n
(
x
)
=
0
Problem 3
1
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Find all matrix in $\mathbb{F}_{13}$
Let
F
13
=
0
‾
,
1
‾
,
⋯
,
12
‾
\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}
F
13
=
0
,
1
,
⋯
,
12
be the finite field with
13
13
13
elements (with sum and product modulus
13
13
13
). Find how many matrix
A
A
A
of size
5
5
5
x
5
5
5
with entries in
F
13
\mathbb{F}_{13}
F
13
exist such that
A
5
=
I
A^5 = I
A
5
=
I
where
I
I
I
is the identity matrix of order
5
5
5
Problem 1
1
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A nice limit
Let
R
>
0
R > 0
R
>
0
, be an integer, and let
n
(
R
)
n(R)
n
(
R
)
be the number um triples
(
x
,
y
,
z
)
∈
Z
3
(x, y, z) \in \mathbb{Z}^3
(
x
,
y
,
z
)
∈
Z
3
such that
2
x
2
+
3
y
2
+
5
z
2
=
R
2x^2+3y^2+5z^2 = R
2
x
2
+
3
y
2
+
5
z
2
=
R
. What is the value of
lim
R
→
∞
n
(
1
)
+
n
(
2
)
+
⋯
+
n
(
R
)
R
3
/
2
\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}
lim
R
→
∞
R
3/2
n
(
1
)
+
n
(
2
)
+
⋯
+
n
(
R
)
?