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IberoAmerican Olympiad For University Students
2008 IberoAmerican Olympiad For University Students
2008 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
2
1
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Polynomial f s.t. f(x)|f(x^2-1) - OIMU 2008 Problem 2
Prove that for each natural number
n
n
n
there is a polynomial
f
f
f
with real coefficients and degree
n
n
n
such that
p
(
x
)
=
f
(
x
2
−
1
)
p(x)=f(x^2-1)
p
(
x
)
=
f
(
x
2
−
1
)
is divisible by
f
(
x
)
f(x)
f
(
x
)
over the ring
R
[
x
]
\mathbb{R}[x]
R
[
x
]
.
3
1
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x+1/x^x<2 - OIMU 2008 Problem 3
Prove that
x
+
1
x
x
<
2
x+\frac{1}{x^x}<2
x
+
x
x
1
<
2
for
0
<
x
<
1
0<x<1
0
<
x
<
1
.
7
1
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|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7
Let
A
A
A
be an abelian additive group such that all nonzero elements have infinite order and for each prime number
p
p
p
we have the inequality
∣
A
/
p
A
∣
≤
p
|A/pA|\leq p
∣
A
/
p
A
∣
≤
p
, where
p
A
=
{
p
a
∣
a
∈
A
}
pA = \{pa |a \in A\}
p
A
=
{
p
a
∣
a
∈
A
}
,
p
a
=
a
+
a
+
⋯
+
a
pa = a+a+\cdots+a
p
a
=
a
+
a
+
⋯
+
a
(where the sum has
p
p
p
summands) and
∣
A
/
p
A
∣
|A/pA|
∣
A
/
p
A
∣
is the order of the quotient group
A
/
p
A
A/pA
A
/
p
A
(the index of the subgroup
p
A
pA
p
A
).Prove that each subgroup of
A
A
A
of finite index is isomorphic to
A
A
A
.
6
1
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A^2+B^2=C^2 and A^4+B^4=C^4 - OIMU 2008 Problem 6
a) Determine if there are matrices
A
,
B
,
C
∈
S
L
2
(
Z
)
A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})
A
,
B
,
C
∈
SL
2
(
Z
)
such that
A
2
+
B
2
=
C
2
A^2+B^2=C^2
A
2
+
B
2
=
C
2
.b) Determine if there are matrices
A
,
B
,
C
∈
S
L
2
(
Z
)
A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})
A
,
B
,
C
∈
SL
2
(
Z
)
such that
A
4
+
B
4
=
C
4
A^4+B^4=C^4
A
4
+
B
4
=
C
4
.Note: The notation
A
∈
S
L
2
(
Z
)
A\in \mathrm{SL}_{2}(\mathbb{Z})
A
∈
SL
2
(
Z
)
means that
A
A
A
is a
2
×
2
2\times 2
2
×
2
matrix with integer entries and
det
A
=
1
\det A=1
det
A
=
1
.
5
1
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n=(\sum ai^2)(\sum bi^2)-(\sum aibi)^2 - OIMU 2008 Problem 5
Find all positive integers
n
n
n
such that there are positive integers
a
1
,
⋯
,
a
n
,
b
1
,
⋯
,
b
n
a_1,\cdots,a_n, b_1,\cdots,b_n
a
1
,
⋯
,
a
n
,
b
1
,
⋯
,
b
n
that satisfy
(
a
1
2
+
⋯
+
a
n
2
)
(
b
1
2
+
⋯
+
b
n
2
)
−
(
a
1
b
1
+
⋯
+
a
n
b
n
)
2
=
n
(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n
(
a
1
2
+
⋯
+
a
n
2
)
(
b
1
2
+
⋯
+
b
n
2
)
−
(
a
1
b
1
+
⋯
+
a
n
b
n
)
2
=
n
4
1
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AC,BC tangent to parabola - OIMU 2008 Problem 4
Two vertices
A
,
B
A,B
A
,
B
of a triangle
A
B
C
ABC
A
BC
are located on a parabola
y
=
a
x
2
+
b
x
+
c
y=ax^2 + bx + c
y
=
a
x
2
+
b
x
+
c
with
a
>
0
a>0
a
>
0
in such a way that the sides
A
C
,
B
C
AC,BC
A
C
,
BC
are tangent to the parabola. Let
m
c
m_c
m
c
be the length of the median
C
C
1
CC_1
C
C
1
of triangle
A
B
C
ABC
A
BC
and
S
S
S
be the area of triangle
A
B
C
ABC
A
BC
. Find
S
2
m
c
3
\frac{S^2}{m_c^3}
m
c
3
S
2
1
1
Hide problems
Removing digits preserves n|b - OIMU 2008 Problem 1
Let
n
n
n
be a positive integer that is not divisible by either
2
2
2
or
5
5
5
. In the decimal expansion of
1
n
=
0.
a
1
a
2
a
3
⋯
\frac{1}{n}= 0.a_1a_2a_3\cdots
n
1
=
0.
a
1
a
2
a
3
⋯
a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to
a
b
\frac{a}{b}
b
a
for some integers
a
,
b
a,b
a
,
b
. Prove that
b
b
b
is divisible by
n
n
n
.