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Undergraduate contests
IberoAmerican Olympiad For University Students
2007 IberoAmerican Olympiad For University Students
2007 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
7
1
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Consecutive numbers heights
The height of a positive integer is defined as being the fraction
s
(
a
)
a
\frac{s(a)}{a}
a
s
(
a
)
, where
s
(
a
)
s(a)
s
(
a
)
is the sum of all the positive divisors of
a
a
a
. Show that for every pair of positive integers
N
,
k
N,k
N
,
k
there is a positive integer
b
b
b
such that the height of each of
b
,
b
+
1
,
⋯
,
b
+
k
b,b+1,\cdots,b+k
b
,
b
+
1
,
⋯
,
b
+
k
is greater than
N
N
N
.
6
1
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Finite and cyclic colombian group
Let
F
F
F
be a field whose characteristic is not
2
2
2
, let
F
∗
=
F
∖
{
0
}
F^*=F\setminus\left\{0\right\}
F
∗
=
F
∖
{
0
}
be its multiplicative group and let
T
T
T
be the subgroup of
F
∗
F^*
F
∗
constituted by its finite order elements. Prove that if
T
T
T
is finite, then
T
T
T
is cyclic and its order is even.
5
1
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Complex polynomials composition
Determine all pairs of polynomials
f
,
g
∈
C
[
x
]
f,g\in\mathbb{C}[x]
f
,
g
∈
C
[
x
]
with complex coefficients such that the following equalities hold for all
x
∈
C
x\in\mathbb{C}
x
∈
C
:
f
(
f
(
x
)
)
−
g
(
g
(
x
)
)
=
1
+
i
f(f(x))-g(g(x))=1+i
f
(
f
(
x
))
−
g
(
g
(
x
))
=
1
+
i
f
(
g
(
x
)
)
−
g
(
f
(
x
)
)
=
1
−
i
f(g(x))-g(f(x))=1-i
f
(
g
(
x
))
−
g
(
f
(
x
))
=
1
−
i
4
1
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Good numbers
Consider an infinite sequence
a
1
,
a
2
,
⋯
a_1,a_2,\cdots
a
1
,
a
2
,
⋯
whose terms all belong to
{
1
,
2
}
\left\{1,2\right\}
{
1
,
2
}
. A positive integer with
n
n
n
digits is said to be good if its decimal representation has the form
a
r
a
r
+
1
⋯
a
r
+
(
n
−
1
)
a_ra_{r+1}\cdots a_{r+(n-1)}
a
r
a
r
+
1
⋯
a
r
+
(
n
−
1
)
, for some positive integer
r
r
r
. Suppose that there are at least
2008
2008
2008
good numbers with a million digits. Prove that there are at least
2008
2008
2008
good numbers with
2007
2007
2007
digits.
3
1
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Periodic integral
Let
f
:
R
→
R
+
f:\mathbb{R}\to\mathbb{R}^+
f
:
R
→
R
+
be a continuous and periodic function. Prove that for all
α
∈
R
\alpha\in\mathbb{R}
α
∈
R
the following inequality holds:
∫
0
T
f
(
x
)
f
(
x
+
α
)
d
x
≥
T
\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T
∫
0
T
f
(
x
+
α
)
f
(
x
)
d
x
≥
T
,where
T
T
T
is the period of
f
(
x
)
f(x)
f
(
x
)
.
2
1
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Power colombian inequality
Prove that for all positive integers
n
n
n
and for all real numbers
x
x
x
such that
0
≤
x
≤
1
0\le x\le1
0
≤
x
≤
1
, the following inequality holds:
(
1
−
x
+
x
2
2
)
n
−
(
1
−
x
)
n
≤
x
2
\left(1-x+\frac{x^2}{2}\right)^n-(1-x)^n\le\frac{x}{2}
(
1
−
x
+
2
x
2
)
n
−
(
1
−
x
)
n
≤
2
x
.
1
1
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Linear independence preservation
For each pair of integers
(
i
,
k
)
(i,k)
(
i
,
k
)
such that
1
≤
i
≤
k
1\le i\le k
1
≤
i
≤
k
, the linear transformation
P
i
,
k
:
R
k
→
R
k
P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k
P
i
,
k
:
R
k
→
R
k
is defined as:
P
i
,
k
(
a
1
,
⋯
,
a
i
−
1
,
a
i
,
a
i
+
1
,
⋯
,
a
k
)
=
(
a
1
,
⋯
,
a
i
−
1
,
0
,
a
i
+
1
,
⋯
,
a
k
)
P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)
P
i
,
k
(
a
1
,
⋯
,
a
i
−
1
,
a
i
,
a
i
+
1
,
⋯
,
a
k
)
=
(
a
1
,
⋯
,
a
i
−
1
,
0
,
a
i
+
1
,
⋯
,
a
k
)
Prove that for all
n
≥
2
n\ge2
n
≥
2
and for every set of
n
−
1
n-1
n
−
1
linearly independent vectors
v
1
,
⋯
,
v
n
−
1
v_1,\cdots,v_{n-1}
v
1
,
⋯
,
v
n
−
1
in
R
n
\mathbb{R}^n
R
n
, there is an integer
k
k
k
such that
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
and such that the vectors
P
k
,
n
(
v
1
)
,
⋯
,
P
k
,
n
(
v
n
−
1
)
P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})
P
k
,
n
(
v
1
)
,
⋯
,
P
k
,
n
(
v
n
−
1
)
are linearly independent.