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Undergraduate contests
Brazil Undergrad MO
2016 Brazil Undergrad MO
2016 Brazil Undergrad MO
Part of
Brazil Undergrad MO
Subcontests
(6)
6
1
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Inequalities and pretty functions
Let it
C
,
D
>
0
C,D > 0
C
,
D
>
0
. We call a function
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
pretty if
f
f
f
is a
C
2
C^2
C
2
-class,
∣
x
3
f
(
x
)
∣
≤
C
|x^3f(x)| \leq C
∣
x
3
f
(
x
)
∣
≤
C
and
∣
x
f
′
′
(
x
)
∣
≤
D
|xf''(x)| \leq D
∣
x
f
′′
(
x
)
∣
≤
D
.[list='i'] [*] Show that if
f
f
f
is pretty, then, given
ϵ
≥
0
\epsilon \geq 0
ϵ
≥
0
, there is a
x
0
≥
0
x_0 \geq 0
x
0
≥
0
such that for every
x
x
x
with
∣
x
∣
≥
x
0
|x| \geq x_0
∣
x
∣
≥
x
0
, we have
∣
x
2
f
′
(
x
)
∣
<
2
C
D
+
ϵ
|x^2f'(x)| < \sqrt{2CD}+\epsilon
∣
x
2
f
′
(
x
)
∣
<
2
C
D
+
ϵ
. [*] Show that if
0
<
E
<
2
C
D
0 < E < \sqrt{2CD}
0
<
E
<
2
C
D
then there is a pretty function
f
f
f
such that for every
x
0
≥
0
x_0 \geq 0
x
0
≥
0
there is a
x
>
x
0
x > x_0
x
>
x
0
such that
∣
x
2
f
′
(
x
)
∣
>
E
|x^2f'(x)| > E
∣
x
2
f
′
(
x
)
∣
>
E
.
5
1
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Find All Soccer-Ball Polyhedra!
A soccer ball is usually made from a polyhedral fugure model, with two types of faces, hexagons and pentagons, and in every vertex incide three faces - two hexagons and one pentagon.We call a polyhedron soccer-ball if it is similar to the traditional soccer ball, in the following sense: its faces are
m
m
m
-gons or
n
n
n
-gons,
m
≠
n
m \not= n
m
=
n
, and in every vertex incide three faces, two of them being
m
m
m
-gons and the other one being an
n
n
n
-gon.[list='i'] [*] Show that
m
m
m
needs to be even. [*] Find all soccer-ball polyhedra.
4
1
Hide problems
Power of a matrix with integer entries?
Let
A
=
(
<
/
b
r
>
4
−
5
2
5
−
3
<
/
b
r
>
)
A=\left( \begin{array}{cc}</br>4 & -\sqrt{5} \\ 2\sqrt{5} & -3</br>\end{array} \right)
A
=
(
<
/
b
r
>
4
2
5
−
5
−
3
<
/
b
r
>
)
Find all pairs of integers
m
,
n
m,n
m
,
n
with
n
≥
1
n \geq 1
n
≥
1
and
∣
m
∣
≤
n
|m| \leq n
∣
m
∣
≤
n
such as all entries of
A
n
−
(
m
+
n
2
)
A
A^n-(m+n^2)A
A
n
−
(
m
+
n
2
)
A
are integer.
3
1
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Primes and linear recurrences
Let it
k
≥
1
k \geq 1
k
≥
1
be an integer. Define the sequence
(
a
n
)
n
≥
1
(a_n)_{n \geq 1}
(
a
n
)
n
≥
1
by
a
0
=
0
,
a
1
=
1
a_0=0,a_1=1
a
0
=
0
,
a
1
=
1
and
a
n
+
2
=
k
a
n
+
1
+
a
n
a_{n+2} = ka_{n+1}+a_n
a
n
+
2
=
k
a
n
+
1
+
a
n
for
n
≥
0
n \geq 0
n
≥
0
. Let it
p
p
p
an odd prime number. Denote
m
(
p
)
m(p)
m
(
p
)
as the smallest positive integer
m
m
m
such that
p
∣
a
m
p | a_m
p
∣
a
m
. Denote
T
(
p
)
T(p)
T
(
p
)
as the smallest positive integer
T
T
T
such that for every natural
j
j
j
we gave
p
∣
(
a
T
+
j
−
a
j
)
p | (a_{T+j}-a_j)
p
∣
(
a
T
+
j
−
a
j
)
. [list='i'] [*] Show that
T
(
p
)
≤
(
p
−
1
)
⋅
m
(
p
)
T(p) \leq (p-1) \cdot m(p)
T
(
p
)
≤
(
p
−
1
)
⋅
m
(
p
)
. [*] Show that if
T
(
p
)
=
(
p
−
1
)
⋅
m
(
p
)
T(p) = (p-1) \cdot m(p)
T
(
p
)
=
(
p
−
1
)
⋅
m
(
p
)
then
∏
1
≤
j
≤
T
(
p
)
−
1
j
≢
0
(
m
o
d
m
(
p
)
)
a
j
≡
(
−
1
)
m
(
p
)
−
1
(
m
o
d
p
)
\prod_{1 \leq j \leq T(p)-1}^{j \not \equiv 0 \pmod{m(p)}}{a_j} \equiv (-1)^{m(p)-1} \pmod{p}
1
≤
j
≤
T
(
p
)
−
1
∏
j
≡
0
(
mod
m
(
p
))
a
j
≡
(
−
1
)
m
(
p
)
−
1
(
mod
p
)
2
1
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YAFTFP - Yet Another "Find The Functions" Problem
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
x
2
+
y
2
f
(
x
)
)
=
x
f
(
y
)
2
−
f
(
x
)
2
f(x^2+y^2f(x)) = xf(y)^2-f(x)^2
f
(
x
2
+
y
2
f
(
x
))
=
x
f
(
y
)
2
−
f
(
x
)
2
for every
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
1
1
Hide problems
'Amortized' series implies limit of arithmetical mean = 0
Let
(
a
n
)
n
≥
1
(a_n)_{n \geq 1}
(
a
n
)
n
≥
1
s sequence of reals such that
∑
n
≥
1
a
n
n
\sum_{n \geq 1}{\frac{a_n}{n}}
n
≥
1
∑
n
a
n
converges. Show that
lim
n
→
∞
1
n
⋅
∑
k
=
1
n
a
k
=
0
\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0
n
→
∞
lim
n
1
⋅
k
=
1
∑
n
a
k
=
0