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Contests
Undergraduate contests
SEEMOUS
2013 SEEMOUS
2013 SEEMOUS
Part of
SEEMOUS
Subcontests
(4)
Problem 4
1
Hide problems
∃n:A^n=-I_2 implies A^2=-I_2 or A^3=-I_2
Let
A
∈
M
2
(
Q
)
A\in M_2(\mathbb Q)
A
∈
M
2
(
Q
)
such that there is
n
∈
N
,
n
≠
0
n\in\mathbb N,n\ne0
n
∈
N
,
n
=
0
, with
A
n
=
−
I
2
A^n=-I_2
A
n
=
−
I
2
. Prove that either
A
2
=
−
I
2
A^2=-I_2
A
2
=
−
I
2
or
A
3
=
−
I
2
A^3=-I_2
A
3
=
−
I
2
.
Problem 3
1
Hide problems
integral inequality, condition 1≥int|f'(x)|^2 from 0 to 1
Find the maximum value of
∫
0
1
∣
f
′
(
x
)
∣
2
∣
f
(
x
)
∣
1
x
d
x
\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx
∫
0
1
∣
f
′
(
x
)
∣
2
∣
f
(
x
)
∣
x
1
d
x
over all continuously differentiable functions
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
with
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
∫
0
1
∣
f
′
(
x
)
∣
2
d
x
≤
1.
\int^1_0|f'(x)|^2dx\le1.
∫
0
1
∣
f
′
(
x
)
∣
2
d
x
≤
1.
Problem 2
1
Hide problems
over M2(C), M^2+N^2=0 and MN+NM=I, existence of matrix A
Let
M
,
N
∈
M
2
(
C
)
M,N\in M_2(\mathbb C)
M
,
N
∈
M
2
(
C
)
be two nonzero matrices such that
M
2
=
N
2
=
0
2
and
M
N
+
N
M
=
I
2
M^2=N^2=0_2\text{ and }MN+NM=I_2
M
2
=
N
2
=
0
2
and
MN
+
NM
=
I
2
where
0
2
0_2
0
2
is the
2
×
2
2\times2
2
×
2
zero matrix and
I
2
I_2
I
2
the
2
×
2
2\times2
2
×
2
unit matrix. Prove that there is an invertible matrix
A
∈
M
2
(
C
)
A\in M_2(\mathbb C)
A
∈
M
2
(
C
)
such that
M
=
A
(
0
1
0
0
)
A
−
1
and
N
=
A
(
0
0
1
0
)
A
−
1
.
M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.
M
=
A
(
0
0
1
0
)
A
−
1
and
N
=
A
(
0
1
0
0
)
A
−
1
.
Problem 1
1
Hide problems
functions satisfing integral equation
Find all continuous functions
f
:
[
1
,
8
]
→
R
f:[1,8]\to\mathbb R
f
:
[
1
,
8
]
→
R
, such that
∫
1
2
f
(
t
3
)
2
d
t
+
2
∫
1
2
f
(
t
3
)
d
t
=
2
3
∫
1
8
f
(
t
)
d
t
−
∫
1
2
(
t
2
−
1
)
2
d
t
.
\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.
∫
1
2
f
(
t
3
)
2
d
t
+
2
∫
1
2
f
(
t
3
)
d
t
=
3
2
∫
1
8
f
(
t
)
d
t
−
∫
1
2
(
t
2
−
1
)
2
d
t
.