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Contests
Undergraduate contests
IberoAmerican Olympiad For University Students
2010 IberoAmerican Olympiad For University Students
2010 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
7
1
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Strange power sum
(a) Prove that, for any positive integers
m
≤
ℓ
m\le \ell
m
≤
ℓ
given, there is a positive integer
n
n
n
and positive integers
x
1
,
⋯
,
x
n
,
y
1
,
⋯
,
y
n
x_1,\cdots,x_n,y_1,\cdots,y_n
x
1
,
⋯
,
x
n
,
y
1
,
⋯
,
y
n
such that the equality
∑
i
=
1
n
x
i
k
=
∑
i
=
1
n
y
i
k
\sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k
i
=
1
∑
n
x
i
k
=
i
=
1
∑
n
y
i
k
holds for every
k
=
1
,
2
,
⋯
,
m
−
1
,
m
+
1
,
⋯
,
ℓ
k=1,2,\cdots,m-1,m+1,\cdots,\ell
k
=
1
,
2
,
⋯
,
m
−
1
,
m
+
1
,
⋯
,
ℓ
, but does not hold for
k
=
m
k=m
k
=
m
.(b) Prove that there is a solution of the problem, where all numbers
x
1
,
⋯
,
x
n
,
y
1
,
⋯
,
y
n
x_1,\cdots,x_n,y_1,\cdots,y_n
x
1
,
⋯
,
x
n
,
y
1
,
⋯
,
y
n
are distinct.Proposed by Ilya Bogdanov and Géza Kós.
6
1
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Quartic polynomial prime divisors
Prove that, for all integer
a
>
1
a>1
a
>
1
, the prime divisors of
5
a
4
−
5
a
2
+
1
5a^4-5a^2+1
5
a
4
−
5
a
2
+
1
have the form
20
k
±
1
,
k
∈
Z
20k\pm1,k\in\mathbb{Z}
20
k
±
1
,
k
∈
Z
.Proposed by Géza Kós.
5
1
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Roots of identity matrix with the same trace
Let
A
,
B
A,B
A
,
B
be matrices of dimension
2010
×
2010
2010\times2010
2010
×
2010
which commute and have real entries, such that
A
2010
=
B
2010
=
I
A^{2010}=B^{2010}=I
A
2010
=
B
2010
=
I
, where
I
I
I
is the identity matrix. Prove that if
tr
(
A
B
)
=
2010
\operatorname{tr}(AB)=2010
tr
(
A
B
)
=
2010
, then
tr
(
A
)
=
tr
(
B
)
\operatorname{tr}(A)=\operatorname{tr}(B)
tr
(
A
)
=
tr
(
B
)
.
4
1
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No consecutive zero coefficients
Let
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
be a monic polynomial of degree
n
>
2
n>2
n
>
2
, with real coefficients and all its roots real and different from zero. Prove that for all
k
=
0
,
1
,
2
,
⋯
,
n
−
2
k=0,1,2,\cdots,n-2
k
=
0
,
1
,
2
,
⋯
,
n
−
2
, at least one of the coefficients
a
k
,
a
k
+
1
a_k,a_{k+1}
a
k
,
a
k
+
1
is different from zero.
3
1
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Strange student's addition
A student adds up rational fractions incorrectly:\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y} (\star) Despite that, he sometimes obtains correct results. For a given fraction
a
b
,
a
,
b
∈
Z
,
b
>
0
\frac{a}{b},a,b\in\mathbb{Z},b>0
b
a
,
a
,
b
∈
Z
,
b
>
0
, find all fractions
x
y
,
x
,
y
∈
Z
,
y
>
0
\frac{x}{y},x,y\in\mathbb{Z},y>0
y
x
,
x
,
y
∈
Z
,
y
>
0
such that the result obtained by
(
⋆
)
(\star)
(
⋆
)
is correct.
2
1
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Power sine series
Calculate the sum of the series
∑
−
∞
∞
sin
3
3
k
3
k
\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}
∑
−
∞
∞
3
k
s
i
n
3
3
k
.
1
1
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Function from triangles into real numbers
Let
f
:
S
→
R
f:S\to\mathbb{R}
f
:
S
→
R
be the function from the set of all right triangles into the set of real numbers, defined by
f
(
Δ
A
B
C
)
=
h
r
f(\Delta ABC)=\frac{h}{r}
f
(
Δ
A
BC
)
=
r
h
, where
h
h
h
is the height with respect to the hypotenuse and
r
r
r
is the inscribed circle's radius. Find the image,
I
m
(
f
)
Im(f)
I
m
(
f
)
, of the function.