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IberoAmerican Olympiad For University Students
2005 IberoAmerican Olympiad For University Students
2005 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
7
1
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Polynomial has all roots real- OIMU 2005 Problem 7
Prove that for any integers
n
,
p
n,p
n
,
p
,
0
<
n
≤
p
0<n\leq p
0
<
n
≤
p
, all the roots of the polynomial below are real:
P
n
,
p
(
x
)
=
∑
j
=
0
n
(
p
j
)
(
p
n
−
j
)
x
j
P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j
P
n
,
p
(
x
)
=
j
=
0
∑
n
(
j
p
)
(
n
−
j
p
)
x
j
6
1
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Totally convex => real analytic - OIMU 2005 Problem 6
A smooth function
f
:
I
→
R
f:I\to \mathbb{R}
f
:
I
→
R
is said to be totally convex if
(
−
1
)
k
f
(
k
)
(
t
)
>
0
(-1)^k f^{(k)}(t) > 0
(
−
1
)
k
f
(
k
)
(
t
)
>
0
for all
t
∈
I
t\in I
t
∈
I
and every integer
k
>
0
k>0
k
>
0
(here
I
I
I
is an open interval).Prove that every totally convex function
f
:
(
0
,
+
∞
)
→
R
f:(0,+\infty)\to \mathbb{R}
f
:
(
0
,
+
∞
)
→
R
is real analytic.Note: A function
f
:
I
→
R
f:I\to \mathbb{R}
f
:
I
→
R
is said to be smooth if for every positive integer
k
k
k
the derivative of order
k
k
k
of
f
f
f
is well defined and continuous over
R
\mathbb{R}
R
. A smooth function
f
:
I
→
R
f:I\to \mathbb{R}
f
:
I
→
R
is said to be real analytic if for every
t
∈
I
t\in I
t
∈
I
there exists
ϵ
>
0
\epsilon> 0
ϵ
>
0
such that for all real numbers
h
h
h
with
∣
h
∣
<
ϵ
|h|<\epsilon
∣
h
∣
<
ϵ
the Taylor series
∑
k
≥
0
f
(
k
)
(
t
)
k
!
h
k
\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k
k
≥
0
∑
k
!
f
(
k
)
(
t
)
h
k
converges and is equal to
f
(
t
+
h
)
f(t+h)
f
(
t
+
h
)
.
5
1
Hide problems
Game with base 2 representation - OIMU 2005 Problem 5
Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says
0
0
0
. In each turn except the first the possible moves are determined from the previous number
n
n
n
in the following way: write
n
=
∑
m
∈
O
n
2
m
;
n =\sum_{m\in O_n}2^m;
n
=
m
∈
O
n
∑
2
m
;
the valid numbers are the elements
m
m
m
of
O
n
O_n
O
n
. That way, for example, after Arnaldo says
42
=
2
5
+
2
3
+
2
1
42= 2^5 + 2^3 + 2^1
42
=
2
5
+
2
3
+
2
1
, Bernaldo must respond with
5
5
5
,
3
3
3
or
1
1
1
.We define the sets
A
,
B
⊂
N
A,B\subset \mathbb{N}
A
,
B
⊂
N
in the following way. We have
n
∈
A
n\in A
n
∈
A
iff Arnaldo, saying
n
n
n
in his first turn, has a winning strategy; analogously, we have
n
∈
B
n\in B
n
∈
B
iff Bernaldo has a winning strategy if Arnaldo says
n
n
n
during his first turn. This way,
A
=
{
0
,
2
,
8
,
10
,
⋯
}
,
B
=
{
1
,
3
,
4
,
5
,
6
,
7
,
9
,
⋯
}
A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}
A
=
{
0
,
2
,
8
,
10
,
⋯
}
,
B
=
{
1
,
3
,
4
,
5
,
6
,
7
,
9
,
⋯
}
Define
f
:
N
→
N
f:\mathbb{N}\to \mathbb{N}
f
:
N
→
N
by
f
(
n
)
=
∣
A
∩
{
0
,
1
,
⋯
,
n
−
1
}
∣
f(n)=|A\cap \{0,1,\cdots,n-1\}|
f
(
n
)
=
∣
A
∩
{
0
,
1
,
⋯
,
n
−
1
}
∣
. For example,
f
(
8
)
=
2
f(8) = 2
f
(
8
)
=
2
and
f
(
11
)
=
4
f(11)=4
f
(
11
)
=
4
. Find
lim
n
→
∞
f
(
n
)
log
(
n
)
2005
n
\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}
n
→
∞
lim
n
f
(
n
)
lo
g
(
n
)
2005
4
1
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Locus is equilateral hyperbola - OIMU 2005 Problem 4
A variable tangent
t
t
t
to the circle
C
1
C_1
C
1
, of radius
r
1
r_1
r
1
, intersects the circle
C
2
C_2
C
2
, of radius
r
2
r_2
r
2
in
A
A
A
and
B
B
B
. The tangents to
C
2
C_2
C
2
through
A
A
A
and
B
B
B
intersect in
P
P
P
. Find, as a function of
r
1
r_1
r
1
and
r
2
r_2
r
2
, the distance between the centers of
C
1
C_1
C
1
and
C
2
C_2
C
2
such that the locus of
P
P
P
when
t
t
t
varies is contained in an equilateral hyperbola.Note: A hyperbola is said to be equilateral if its asymptotes are perpendicular.
3
1
Hide problems
Limit of recursive sequence of points - OIMU 2005 Problem 3
Consider the sequence defined recursively by
(
x
1
,
y
1
)
=
(
0
,
0
)
(x_1,y_1)=(0,0)
(
x
1
,
y
1
)
=
(
0
,
0
)
,
(
x
n
+
1
,
y
n
+
1
)
=
(
(
1
−
2
n
)
x
n
−
1
n
y
n
+
4
n
,
(
1
−
1
n
)
y
n
−
1
n
x
n
+
3
n
)
(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)
(
x
n
+
1
,
y
n
+
1
)
=
(
(
1
−
n
2
)
x
n
−
n
1
y
n
+
n
4
,
(
1
−
n
1
)
y
n
−
n
1
x
n
+
n
3
)
.Find
lim
n
→
∞
(
x
n
,
y
n
)
\lim_{n\to \infty}(x_n,y_n)
lim
n
→
∞
(
x
n
,
y
n
)
.
2
1
Hide problems
A^3=-I, BA^2+BA=C^6+C+I, C symmetric - OIMU 2005 Problem 2
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be real square matrices of order
n
n
n
such that
A
3
=
−
I
A^3=-I
A
3
=
−
I
,
B
A
2
+
B
A
=
C
6
+
C
+
I
BA^2+BA=C^6+C+I
B
A
2
+
B
A
=
C
6
+
C
+
I
and
C
C
C
is symmetric. Is it possible that
n
=
2005
n=2005
n
=
2005
?
1
1
Hide problems
Degree of polynomial recursion - OIMU 2005 Problem 1
Let
P
(
x
,
y
)
=
(
x
2
y
3
,
x
3
y
5
)
P(x,y)=(x^2y^3,x^3y^5)
P
(
x
,
y
)
=
(
x
2
y
3
,
x
3
y
5
)
,
P
1
=
P
P^1=P
P
1
=
P
and
P
n
+
1
=
P
∘
P
n
P^{n+1}=P\circ P^n
P
n
+
1
=
P
∘
P
n
. Also, let
p
n
(
x
)
p_n(x)
p
n
(
x
)
be the first coordinate of
P
n
(
x
,
x
)
P^n(x,x)
P
n
(
x
,
x
)
, and
f
(
n
)
f(n)
f
(
n
)
be the degree of
p
n
(
x
)
p_n(x)
p
n
(
x
)
. Find
lim
n
→
∞
f
(
n
)
1
/
n
\lim_{n\to\infty}f(n)^{1/n}
n
→
∞
lim
f
(
n
)
1/
n