MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexican University Math Olympiad
2024 Mexican University Math Olympiad
2024 Mexican University Math Olympiad
Part of
Mexican University Math Olympiad
Subcontests
(6)
6
1
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Derivative and bounds of a monic polynomial
Let
p
p
p
be a monic polynomial with all distinct real roots. Show that there exists
K
K
K
such that
(
p
(
x
)
2
)
′
′
≤
K
(
p
′
(
x
)
)
2
.
(p(x)^2)'' \leq K(p'(x))^2.
(
p
(
x
)
2
)
′′
≤
K
(
p
′
(
x
)
)
2
.
5
1
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Permutation and sum of sequences
Consider two finite sequences of real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
n
b_1, b_2, \dots, b_n
b
1
,
b
2
,
…
,
b
n
. Let
α
(
x
)
=
#
{
i
∣
a
i
=
x
}
\alpha(x) = \#\{i | a_i = x \}
α
(
x
)
=
#
{
i
∣
a
i
=
x
}
and
β
(
x
)
=
#
{
i
∣
b
i
=
−
x
}
\beta(x) = \#\{i | b_i = -x \}
β
(
x
)
=
#
{
i
∣
b
i
=
−
x
}
. Prove that there exists a permutation
σ
∈
S
n
\sigma \in S_n
σ
∈
S
n
(the symmetric group of
n
n
n
elements) such that
a
σ
(
i
)
+
b
i
≠
0
a_{\sigma(i)} + b_i \neq 0
a
σ
(
i
)
+
b
i
=
0
for all
i
=
1
,
…
,
n
i = 1, \dots, n
i
=
1
,
…
,
n
if and only if
α
(
x
)
+
β
(
x
)
≤
n
\alpha(x) + \beta(x) \leq n
α
(
x
)
+
β
(
x
)
≤
n
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
.
4
1
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Limit of matrix entries
Given
b
>
0
b > 0
b
>
0
, consider the following matrix:
B
=
(
b
b
2
b
2
b
3
)
B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix}
B
=
(
b
b
2
b
2
b
3
)
Denote by
e
i
e_i
e
i
the top left entry of
B
i
B^i
B
i
. Prove that the following limit exists and calculate its value:
lim
i
→
∞
e
i
.
\lim_{i \to \infty} \sqrt{e_i}.
i
→
∞
lim
e
i
.
3
1
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Multiplicative function in the unit disk
Consider a multiplicative function
f
f
f
from the positive integers to the unit disk centered at the origin, that is,
f
:
Z
+
→
D
2
⊆
C
f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C}
f
:
Z
+
→
D
2
⊆
C
such that
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn) = f(m)f(n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
. Prove that for every
ϵ
>
0
\epsilon > 0
ϵ
>
0
and every integer
k
>
0
k > 0
k
>
0
, there exist
k
k
k
distinct positive integers
a
1
,
a
2
,
…
,
a
k
a_1, a_2, \dots, a_k
a
1
,
a
2
,
…
,
a
k
such that
gcd
(
a
1
,
a
2
,
…
,
a
k
)
=
k
\text{gcd}(a_1, a_2, \dots, a_k) = k
gcd
(
a
1
,
a
2
,
…
,
a
k
)
=
k
and
d
(
f
(
a
i
)
,
f
(
a
j
)
)
<
ϵ
d(f(a_i), f(a_j)) < \epsilon
d
(
f
(
a
i
)
,
f
(
a
j
))
<
ϵ
for all
i
,
j
=
1
,
…
,
k
i, j = 1, \dots, k
i
,
j
=
1
,
…
,
k
.
2
1
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Complex matrices and polynomial
Let
A
A
A
and
B
B
B
be two square matrices with complex entries such that
A
+
B
=
A
B
A + B = AB
A
+
B
=
A
B
,
A
=
A
∗
A = A^*
A
=
A
∗
, and
A
A
A
has all distinct eigenvalues. Prove that there exists a polynomial
P
P
P
with complex coefficients such that
P
(
A
)
=
B
P(A) = B
P
(
A
)
=
B
.
1
1
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Equation with primes and perfect square
Let
x
x
x
,
y
y
y
,
p
p
p
be positive integers that satisfy the equation
x
4
=
p
+
9
y
4
x^4 = p + 9y^4
x
4
=
p
+
9
y
4
, where
p
p
p
is a prime number. Show that
p
2
−
1
3
\frac{p^2 - 1}{3}
3
p
2
−
1
is a perfect square and a multiple of 16.