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Problems
Contests
National and Regional Contests
Korea Contests
South Korea USCM
2016 Korea USCM
2016 Korea USCM
Part of
South Korea USCM
Subcontests
(8)
8
1
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equivalent condition to nilpotent
For a
n
×
n
n\times n
n
×
n
complex valued matrix
A
A
A
, show that the following two conditions are equivalent. (i) There exists a
n
×
n
n\times n
n
×
n
complex valued matrix
B
B
B
such that
A
B
−
B
A
=
A
AB-BA=A
A
B
−
B
A
=
A
. (ii) There exists a positive integer
k
k
k
such that
A
k
=
O
A^k = O
A
k
=
O
. (
O
O
O
is the zero matrix.)
7
1
Hide problems
Cauchy like integral inequality
M
M
M
is a postive real and
f
:
[
0
,
∞
)
→
[
0
,
M
]
f:[0,\infty)\to[0,M]
f
:
[
0
,
∞
)
→
[
0
,
M
]
is a continuous function such that
∫
0
∞
(
1
+
x
)
f
(
x
)
d
x
<
∞
\int_0^\infty (1+x)f(x) dx<\infty
∫
0
∞
(
1
+
x
)
f
(
x
)
d
x
<
∞
Then, prove the following inequality.
(
∫
0
∞
f
(
x
)
d
x
)
2
≤
4
M
∫
0
∞
x
f
(
x
)
d
x
\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx
(
∫
0
∞
f
(
x
)
d
x
)
2
≤
4
M
∫
0
∞
x
f
(
x
)
d
x
(@below, Thank you. I fixed.)
6
1
Hide problems
Prove that two 2*2 matrices have common eigenvector
A
A
A
and
B
B
B
are
2
×
2
2\times 2
2
×
2
real valued matrices satisfying \det A = \det B = 1, \text{tr}(A)>2, \text{tr}(B)>2, \text{tr}(ABA^{-1}B^{-1}) = 2 Prove that
A
A
A
and
B
B
B
have a common eigenvector.
5
1
Hide problems
infty norm and -infty norm
For
f
(
x
)
=
cos
(
3
3
π
8
(
x
−
x
3
)
)
f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)
f
(
x
)
=
cos
(
8
3
3
π
(
x
−
x
3
)
)
, find the value of
lim
t
→
∞
(
∫
0
1
f
(
x
)
t
d
x
)
1
t
+
lim
t
→
−
∞
(
∫
0
1
f
(
x
)
t
d
x
)
1
t
\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t}
t
→
∞
lim
(
∫
0
1
f
(
x
)
t
d
x
)
t
1
+
t
→
−
∞
lim
(
∫
0
1
f
(
x
)
t
d
x
)
t
1
4
1
Hide problems
easy gronwall type inequality
Suppose a continuous function
f
:
[
−
π
4
,
π
4
]
→
[
−
1
,
1
]
f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]
f
:
[
−
4
π
,
4
π
]
→
[
−
1
,
1
]
and differentiable on
(
−
π
4
,
π
4
)
(-\frac{\pi}{4},\frac{\pi}{4})
(
−
4
π
,
4
π
)
. Then, there exists a point
x
0
∈
(
−
π
4
,
π
4
)
x_0\in (-\frac{\pi}{4},\frac{\pi}{4})
x
0
∈
(
−
4
π
,
4
π
)
such that
∣
f
′
(
x
0
)
∣
≤
1
+
f
(
x
0
)
2
|f'(x_0)|\leq 1+f(x_0)^2
∣
f
′
(
x
0
)
∣
≤
1
+
f
(
x
0
)
2
3
1
Hide problems
Matrix of the form 1+AA^T
Given positive integers
m
,
n
m,n
m
,
n
and a
m
×
n
m\times n
m
×
n
matrix
A
A
A
with real entries. (1) Show that matrices
X
=
I
m
+
A
A
T
X = I_m + AA^T
X
=
I
m
+
A
A
T
and
Y
=
I
n
+
A
T
A
Y = I_n + A^T A
Y
=
I
n
+
A
T
A
are invertible. (
I
l
I_l
I
l
is the
l
×
l
l\times l
l
×
l
unit matrix.) (2) Evaluate the value of
tr
(
X
−
1
)
−
tr
(
Y
−
1
)
\text{tr}(X^{-1}) - \text{tr}(Y^{-1})
tr
(
X
−
1
)
−
tr
(
Y
−
1
)
.
2
1
Hide problems
decreasing seq, upper bound condition on partial sum, show series upper bound
Suppose
{
a
n
}
\{a_n\}
{
a
n
}
is a decreasing sequence of reals and
lim
n
→
∞
a
n
=
0
\lim\limits_{n\to\infty} a_n = 0
n
→
∞
lim
a
n
=
0
. If
S
2
k
−
2
k
a
2
k
≤
1
S_{2^k} - 2^k a_{2^k} \leq 1
S
2
k
−
2
k
a
2
k
≤
1
for any positive integer
k
k
k
, show that
∑
n
=
1
∞
a
n
≤
1
\sum_{n=1}^{\infty} a_n \leq 1
n
=
1
∑
∞
a
n
≤
1
(At here,
S
m
=
∑
n
=
1
m
a
n
S_m = \sum_{n=1}^m a_n
S
m
=
∑
n
=
1
m
a
n
is a partial sum of
{
a
n
}
\{a_n\}
{
a
n
}
.)
1
1
Hide problems
Find limit of summation
Find the following limit.
lim
n
→
∞
1
n
log
(
∑
k
=
2
2
n
k
1
/
n
2
)
\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)
n
→
∞
lim
n
1
lo
g
(
k
=
2
∑
2
n
k
1/
n
2
)