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IberoAmerican Olympiad For University Students
2009 IberoAmerican Olympiad For University Students
2009 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
7
1
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Subgroups subnormal=>Nontrivial center - OIMU 2009 Problem 7
Let
G
G
G
be a group such that every subgroup of
G
G
G
is subnormal. Suppose that there exists
N
N
N
normal subgroup of
G
G
G
such that
Z
(
N
)
Z(N)
Z
(
N
)
is nontrivial and
G
/
N
G/N
G
/
N
is cyclic. Prove that
Z
(
G
)
Z(G)
Z
(
G
)
is nontrivial. (
Z
(
G
)
Z(G)
Z
(
G
)
denotes the center of
G
G
G
).Note: A subgroup
H
H
H
of
G
G
G
is subnormal if there exist subgroups
H
1
,
H
2
,
…
,
H
m
=
G
H_1,H_2,\ldots,H_m=G
H
1
,
H
2
,
…
,
H
m
=
G
of
G
G
G
such that
H
⊲
H
1
⊲
H
2
⊲
…
⊲
H
m
=
G
H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G
H
⊲
H
1
⊲
H
2
⊲
…
⊲
H
m
=
G
(
⊲
\lhd
⊲
denotes normal subgroup).
6
1
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Polynomials in Z[x], |product|=1 - OIMU 2009 Problem 6
Let
α
1
,
…
,
α
d
,
β
1
,
…
,
β
e
∈
C
\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}
α
1
,
…
,
α
d
,
β
1
,
…
,
β
e
∈
C
be such that the polynomials
f
1
(
x
)
=
∏
i
=
1
d
(
x
−
α
i
)
f_1(x) =\prod_{i=1}^d(x-\alpha_i)
f
1
(
x
)
=
∏
i
=
1
d
(
x
−
α
i
)
and
f
2
(
x
)
=
∏
i
=
1
e
(
x
−
β
i
)
f_2(x) =\prod_{i=1}^e(x-\beta_i)
f
2
(
x
)
=
∏
i
=
1
e
(
x
−
β
i
)
have integer coefficients.Suppose that there exist polynomials
g
1
,
g
2
∈
Z
[
x
]
g_1, g_2 \in\mathbb{Z}[x]
g
1
,
g
2
∈
Z
[
x
]
such that
f
1
g
1
+
f
2
g
2
=
1
f_1g_1 +f_2g_2 = 1
f
1
g
1
+
f
2
g
2
=
1
.Prove that \left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j) \right|=1
5
1
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Two binary relations - OIMU 2009 Problem 5
Let
N
\mathbb{N}
N
and
N
∗
\mathbb{N}^*
N
∗
be the sets containing the natural numbers/positive integers respectively.We define a binary relation on
N
\mathbb{N}
N
by
a
∈
ˊ
b
a\acute{\in}b
a
∈
ˊ
b
iff the
a
a
a
-th bit in the binary representation of
b
b
b
is
1
1
1
.We define a binary relation on
N
∗
\mathbb{N}^*
N
∗
by
a
∈
~
b
a\tilde{\in}b
a
∈
~
b
iff
b
b
b
is a multiple of the
a
a
a
-th prime number
p
a
p_a
p
a
.i) Prove that there is no bijection
f
:
N
→
N
∗
f:\mathbb{N}\to \mathbb{N}^*
f
:
N
→
N
∗
such that
a
∈
ˊ
b
⇔
f
(
a
)
∈
~
f
(
b
)
a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)
a
∈
ˊ
b
⇔
f
(
a
)
∈
~
f
(
b
)
. ii) Prove that there is a bijection
g
:
N
→
N
∗
g:\mathbb{N}\to \mathbb{N}^*
g
:
N
→
N
∗
such that
(
a
∈
ˊ
b
∨
b
∈
ˊ
a
)
⇔
(
g
(
a
)
∈
~
g
(
b
)
∨
g
(
b
)
∈
~
g
(
a
)
)
(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))
(
a
∈
ˊ
b
∨
b
∈
ˊ
a
)
⇔
(
g
(
a
)
∈
~
g
(
b
)
∨
g
(
b
)
∈
~
g
(
a
))
.
4
1
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Slippery functions - OIMU 2009 Problem 4
Given two positive integers
m
,
n
m,n
m
,
n
, we say that a function
f
:
[
0
,
m
]
→
R
f : [0,m] \to \mathbb{R}
f
:
[
0
,
m
]
→
R
is
(
m
,
n
)
(m,n)
(
m
,
n
)
-slippery if it has the following properties:i)
f
f
f
is continuous; ii)
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
,
f
(
m
)
=
n
f(m) = n
f
(
m
)
=
n
; iii) If
t
1
,
t
2
∈
[
0
,
m
]
t_1, t_2\in [0,m]
t
1
,
t
2
∈
[
0
,
m
]
with
t
1
<
t
2
t_1 < t_2
t
1
<
t
2
are such that
t
2
−
t
1
∈
Z
t_2-t_1\in \mathbb{Z}
t
2
−
t
1
∈
Z
and
f
(
t
2
)
−
f
(
t
1
)
∈
Z
f(t_2)-f(t_1)\in\mathbb{Z}
f
(
t
2
)
−
f
(
t
1
)
∈
Z
, then
t
2
−
t
1
∈
{
0
,
m
}
t_2-t_1 \in \{0,m\}
t
2
−
t
1
∈
{
0
,
m
}
.Find all the possible values for
m
,
n
m, n
m
,
n
such that there is a function
f
f
f
that is
(
m
,
n
)
(m,n)
(
m
,
n
)
-slippery.
3
1
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Find the maximum of a product - OIMU 2009 Problem 3
Let
a
,
b
,
c
,
d
,
e
∈
R
+
a, b, c, d, e \in \mathbb{R}^+
a
,
b
,
c
,
d
,
e
∈
R
+
and
f
:
{
(
x
,
y
)
∈
(
R
+
)
2
∣
c
−
d
x
−
e
y
>
0
}
→
R
+
f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+
f
:
{(
x
,
y
)
∈
(
R
+
)
2
∣
c
−
d
x
−
ey
>
0
}
→
R
+
be given by
f
(
x
,
y
)
=
(
a
x
)
(
b
y
)
(
c
−
d
x
−
e
y
)
f(x, y) = (ax)(by)(c- dx- ey)
f
(
x
,
y
)
=
(
a
x
)
(
b
y
)
(
c
−
d
x
−
ey
)
. Find the maximum value of
f
f
f
.
2
1
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Linearly independent vectors - OIMU 2009 Problem 2
Let
x
1
,
⋯
,
x
n
x_1,\cdots, x_n
x
1
,
⋯
,
x
n
be nonzero vectors of a vector space
V
V
V
and
φ
:
V
→
V
\varphi:V\to V
φ
:
V
→
V
be a linear transformation such that
φ
x
1
=
x
1
\varphi x_1 = x_1
φ
x
1
=
x
1
,
φ
x
k
=
x
k
−
x
k
−
1
\varphi x_k = x_k - x_{k-1}
φ
x
k
=
x
k
−
x
k
−
1
for
k
=
2
,
3
,
…
,
n
k = 2, 3,\ldots,n
k
=
2
,
3
,
…
,
n
. Prove that the vectors
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
are linearly independent.
1
1
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Triangle cut in similar triangles - OIMU 2009 Problem 1
A line through a vertex of a non-degenerate triangle cuts it in two similar triangles with
3
\sqrt{3}
3
as the ratio between correspondent sides. Find the angles of the given triangle.