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Undergraduate contests
SEEMOUS
2021 SEEMOUS
2021 SEEMOUS
Part of
SEEMOUS
Subcontests
(4)
Problem 4
1
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Convergence of a strange sequence
For
p
∈
R
p \in \mathbb{R}
p
∈
R
, let
(
a
n
)
n
≥
1
(a_n)_{n \ge 1}
(
a
n
)
n
≥
1
be the sequence defined by
a
n
=
1
n
p
∫
0
n
∣
sin
(
π
x
)
∣
x
d
x
.
a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx.
a
n
=
n
p
1
∫
0
n
∣
sin
(
π
x
)
∣
x
d
x
.
Determine all possible values of
p
p
p
for which the series
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
converges.
Problem 3
1
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Normal matrix
Let
A
∈
M
n
(
C
)
A \in \mathcal{M}_n(\mathbb{C})
A
∈
M
n
(
C
)
be a matrix such that
(
A
A
∗
)
2
=
A
∗
A
(AA^*)^2=A^*A
(
A
A
∗
)
2
=
A
∗
A
, where
A
∗
=
(
A
ˉ
)
t
A^*=(\bar{A})^t
A
∗
=
(
A
ˉ
)
t
denotes the Hermitian transpose (i.e., the conjugate transpose) of
A
A
A
. (a) Prove that
A
A
∗
=
A
∗
A
AA^*=A^*A
A
A
∗
=
A
∗
A
. (b) Show that the non-zero eigenvalues of
A
A
A
have modulus one.
Problem 2
1
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Commutative implies non-negative determinant
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer and let
A
∈
M
n
(
R
)
A \in \mathcal{M}_n(\mathbb{R})
A
∈
M
n
(
R
)
be a matrix such that
A
2
=
−
I
n
A^2=-I_n
A
2
=
−
I
n
. If
B
∈
M
n
(
R
)
B \in \mathcal{M}_n(\mathbb{R})
B
∈
M
n
(
R
)
and
A
B
=
B
A
AB = BA
A
B
=
B
A
, prove that
det
B
≥
0
\det B \ge 0
det
B
≥
0
.
Problem 1
1
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Harmonic series revisited
Let
f
:
[
0
,
1
]
→
R
f: [0, 1] \to \mathbb{R}
f
:
[
0
,
1
]
→
R
be a continuous strictly increasing function such that
lim
x
→
0
+
f
(
x
)
x
=
1.
\lim_{x \to 0^+} \frac{f(x)}{x}=1.
x
→
0
+
lim
x
f
(
x
)
=
1.
(a) Prove that the sequence
(
x
n
)
n
≥
1
(x_n)_{n \ge 1}
(
x
n
)
n
≥
1
defined by
x
n
=
f
(
1
1
)
+
f
(
1
2
)
+
⋯
+
f
(
1
n
)
−
∫
1
n
f
(
1
x
)
d
x
x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx
x
n
=
f
(
1
1
)
+
f
(
2
1
)
+
⋯
+
f
(
n
1
)
−
∫
1
n
f
(
x
1
)
d
x
is convergent. (b) Find the limit of the sequence
(
y
n
)
n
≥
1
(y_n)_{n \ge 1}
(
y
n
)
n
≥
1
defined by
y
n
=
f
(
1
n
+
1
)
+
f
(
1
n
+
2
)
+
⋯
+
f
(
1
2021
n
)
.
y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right).
y
n
=
f
(
n
+
1
1
)
+
f
(
n
+
2
1
)
+
⋯
+
f
(
2021
n
1
)
.