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Undergraduate contests
SEEMOUS
2024 SEEMOUS
2024 SEEMOUS
Part of
SEEMOUS
Subcontests
(4)
P4
1
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Necessary condition for symmetric matrix
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
,
n
≥
2
n\geq 2
n
≥
2
. Find all values of
k
∈
N
k\in\mathbb{N}
k
∈
N
,
k
≥
1
k\geq 1
k
≥
1
, for which the following statement holds:
"If
A
∈
M
n
(
C
)
is such that
A
k
A
∗
=
A
, then
A
=
A
∗
."
\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."}
"If
A
∈
M
n
(
C
)
is such that
A
k
A
∗
=
A
, then
A
=
A
∗
."
(here,
A
∗
A^*
A
∗
denotes the conjugate transpose of
A
A
A
).
P3
1
Hide problems
Limits concerning an integral sequence
For every
n
≥
1
n\geq 1
n
≥
1
define
x
n
x_n
x
n
by
x
n
=
∫
0
1
ln
(
1
+
x
+
x
2
+
⋯
+
x
n
)
⋅
ln
1
1
−
x
d
x
.
x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx.
x
n
=
∫
0
1
ln
(
1
+
x
+
x
2
+
⋯
+
x
n
)
⋅
ln
1
−
x
1
d
x
.
a) Show that
x
n
x_n
x
n
is finite for every
n
≥
1
n\geq 1
n
≥
1
and
lim
n
→
∞
x
n
=
2
\lim_{n\rightarrow\infty}x_n=2
lim
n
→
∞
x
n
=
2
. b) Calculate
lim
n
→
∞
n
ln
n
(
2
−
x
n
)
\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n)
lim
n
→
∞
l
n
n
n
(
2
−
x
n
)
.
P2
1
Hide problems
Freshman's dream for matrices
Let
A
,
B
∈
M
n
(
R
)
A,B\in\mathcal{M}_n(\mathbb{R})
A
,
B
∈
M
n
(
R
)
two real, symmetric matrices with nonnegative eigenvalues. Prove that
A
3
+
B
3
=
(
A
+
B
)
3
A^3+B^3=(A+B)^3
A
3
+
B
3
=
(
A
+
B
)
3
if and only if
A
B
=
O
n
AB=O_n
A
B
=
O
n
.
P1
1
Hide problems
Recursive sequence has convergent power series
Let
(
x
n
)
n
≥
1
(x_n)_{n\geq 1}
(
x
n
)
n
≥
1
be the sequence defined by
x
1
∈
(
0
,
1
)
x_1\in (0,1)
x
1
∈
(
0
,
1
)
and
x
n
+
1
=
x
n
−
x
n
2
n
x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}
x
n
+
1
=
x
n
−
n
x
n
2
for all
n
≥
1
n\geq 1
n
≥
1
. Find the values of
α
∈
R
\alpha\in\mathbb{R}
α
∈
R
for which the series
∑
n
=
1
∞
x
n
α
\sum_{n=1}^{\infty}x_n^{\alpha}
∑
n
=
1
∞
x
n
α
is convergent.