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Problems
Contests
National and Regional Contests
Iran Contests
Pre-Preparation Course Examination
2012 Pre-Preparation Course Examination
2012 Pre-Preparation Course Examination
Part of
Pre-Preparation Course Examination
Subcontests
(6)
6
2
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absolutely convergent sequences
Suppose that
a
i
j
a_{ij}
a
ij
are real numbers in such a way that for each
i
i
i
, the series
∑
j
=
1
∞
a
i
j
\sum_{j=1}^{\infty}a_{ij}
∑
j
=
1
∞
a
ij
is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence
{
b
j
}
j
\{b_j\}_j
{
b
j
}
j
we have
lim
i
→
∞
∑
j
=
1
∞
a
i
j
b
j
=
0
\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0
lim
i
→
∞
∑
j
=
1
∞
a
ij
b
j
=
0
. Prove that
lim
i
→
∞
∑
j
=
1
∞
∣
a
i
j
∣
=
0
\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0
lim
i
→
∞
∑
j
=
1
∞
∣
a
ij
∣
=
0
.
Inner product vector space
Suppose that
V
V
V
is a finite dimensional vector space over the real numbers equipped with an inner product and
S
:
V
×
V
⟶
R
S:V\times V \longrightarrow \mathbb R
S
:
V
×
V
⟶
R
is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation
T
:
V
⟶
V
T:V \longrightarrow V
T
:
V
⟶
V
such that
∀
u
,
v
∈
V
:
S
(
u
,
v
)
=
<
u
,
T
(
v
)
>
\forall u,v \in V: S(u,v)=<u,T(v)>
∀
u
,
v
∈
V
:
S
(
u
,
v
)
=<
u
,
T
(
v
)
>
.We know that there always exists
v
∈
V
v\in V
v
∈
V
such that
W
=
<
v
,
T
(
v
)
>
W=<v,T(v)>
W
=<
v
,
T
(
v
)
>
is invariant under
T
T
T
. (it means
T
(
W
)
⊆
W
T(W)\subseteq W
T
(
W
)
⊆
W
). Prove that if
W
W
W
is invariant under
T
T
T
then the following subspace is also invariant under
T
T
T
:
W
⊥
=
{
v
∈
V
:
∀
u
∈
W
<
v
,
u
>
=
0
}
W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}
W
⊥
=
{
v
∈
V
:
∀
u
∈
W
<
v
,
u
>=
0
}
.Prove that if dimension of
V
V
V
is more than
3
3
3
, then there exist a two dimensional subspace
W
W
W
of
V
V
V
such that the volume defined on it by function
S
S
S
is zero!!!!(This is the way that we can define a two dimensional volume for each subspace
V
V
V
. This can be done for volumes of higher dimensions.)
5
2
Hide problems
a relation between a function and it's second derivative
The
2
n
d
2^{nd}
2
n
d
order differentiable function
f
:
R
⟶
R
f:\mathbb R \longrightarrow \mathbb R
f
:
R
⟶
R
is in such a way that for every
x
∈
R
x\in \mathbb R
x
∈
R
we have
f
′
′
(
x
)
+
f
(
x
)
=
0
f''(x)+f(x)=0
f
′′
(
x
)
+
f
(
x
)
=
0
.a) Prove that if in addition,
f
(
0
)
=
f
′
(
0
)
=
0
f(0)=f'(0)=0
f
(
0
)
=
f
′
(
0
)
=
0
, then
f
≡
0
f\equiv 0
f
≡
0
.b) Use the previous part to show that there exist
a
,
b
∈
R
a,b\in \mathbb R
a
,
b
∈
R
such that
f
(
x
)
=
a
sin
x
+
b
cos
x
f(x)=a\sin x+b\cos x
f
(
x
)
=
a
sin
x
+
b
cos
x
.
p(T) is invertible or zero
Suppose that for the linear transformation
T
:
V
⟶
V
T:V \longrightarrow V
T
:
V
⟶
V
where
V
V
V
is a vector space, there is no trivial subspace
W
⊂
V
W\subset V
W
⊂
V
such that
T
(
W
)
⊆
W
T(W)\subseteq W
T
(
W
)
⊆
W
. Prove that for every polynomial
p
(
x
)
p(x)
p
(
x
)
, the transformation
p
(
T
)
p(T)
p
(
T
)
is invertible or zero.
4
2
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each compact set has a compact pre-image
Suppose that
X
X
X
and
Y
Y
Y
are metric spaces and
f
:
X
⟶
Y
f:X \longrightarrow Y
f
:
X
⟶
Y
is a continious function. Also
f
1
:
X
×
R
⟶
Y
×
R
f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R
f
1
:
X
×
R
⟶
Y
×
R
with equation
f
1
(
x
,
t
)
=
(
f
(
x
)
,
t
)
f_1(x,t)=(f(x),t)
f
1
(
x
,
t
)
=
(
f
(
x
)
,
t
)
for all
x
∈
X
x\in X
x
∈
X
and
t
∈
R
t\in \mathbb R
t
∈
R
is a closed function. Prove that for every compact set
K
⊆
Y
K\subseteq Y
K
⊆
Y
, its pre-image
f
p
r
e
(
K
)
f^{pre}(K)
f
p
re
(
K
)
is a compact set in
X
X
X
.
Upper triangular matrix
Prove that these two statements are equivalent for an
n
n
n
dimensional vector space
V
V
V
:
⋅
\cdot
⋅
For the linear transformation
T
:
V
⟶
V
T:V\longrightarrow V
T
:
V
⟶
V
there exists a base for
V
V
V
such that the representation of
T
T
T
in that base is an upper triangular matrix.
⋅
\cdot
⋅
There exist subspaces
{
0
}
⊊
V
1
⊊
.
.
.
⊊
V
n
−
1
⊊
V
\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V
{
0
}
⊊
V
1
⊊
...
⊊
V
n
−
1
⊊
V
such that for all
i
i
i
,
T
(
V
i
)
⊆
V
i
T(V_i)\subseteq V_i
T
(
V
i
)
⊆
V
i
.
3
2
Hide problems
there is no function satisfying the minimum
Consider the set
A
=
{
f
∈
C
1
(
[
−
1
,
1
]
)
:
f
(
−
1
)
=
−
1
,
f
(
1
)
=
1
}
\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}
A
=
{
f
∈
C
1
([
−
1
,
1
])
:
f
(
−
1
)
=
−
1
,
f
(
1
)
=
1
}
.Prove that there is no function in this function space that gives us the minimum of
S
=
∫
−
1
1
x
2
f
′
(
x
)
2
d
x
S=\int_{-1}^1x^2f'(x)^2dx
S
=
∫
−
1
1
x
2
f
′
(
x
)
2
d
x
. What is the infimum of
S
S
S
for the functions of this space?
Two linear transformations
Suppose that
T
,
U
:
V
⟶
V
T,U:V\longrightarrow V
T
,
U
:
V
⟶
V
are two linear transformations on the vector space
V
V
V
such that
T
+
U
T+U
T
+
U
is an invertible transformation. Prove that
T
U
=
U
T
=
0
⇔
rank
T
+
rank
U
=
n
TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n
T
U
=
U
T
=
0
⇔
rank
T
+
rank
U
=
n
.
2
2
Hide problems
convergent sequences
Suppose that
lim
n
→
∞
a
n
=
a
\lim_{n\to \infty} a_n=a
lim
n
→
∞
a
n
=
a
and
lim
n
→
∞
b
n
=
b
\lim_{n\to \infty} b_n=b
lim
n
→
∞
b
n
=
b
. Prove that
lim
n
→
∞
1
n
(
a
1
b
n
+
a
2
b
n
−
1
+
.
.
.
+
a
n
b
1
)
=
a
b
\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab
lim
n
→
∞
n
1
(
a
1
b
n
+
a
2
b
n
−
1
+
...
+
a
n
b
1
)
=
ab
.
Number of subspaces not less than cardinality of the field
Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.
1
2
Hide problems
pre image of a closed set is closed
Suppose that
X
X
X
and
Y
Y
Y
are two metric spaces and
f
:
X
⟶
Y
f:X \longrightarrow Y
f
:
X
⟶
Y
is a continious function. Also for every compact set
K
⊆
Y
K \subseteq Y
K
⊆
Y
, it's pre-image
f
p
r
e
(
K
)
f^{pre}(K)
f
p
re
(
K
)
is a compact set in
X
X
X
. Prove that
f
f
f
is a closed function, i.e for every close set
C
⊆
X
C\subseteq X
C
⊆
X
, it's image
f
(
C
)
f(C)
f
(
C
)
is a closed subset of
Y
Y
Y
.
A subspace is the direct sum of two others
Suppose that
W
,
W
1
W,W_1
W
,
W
1
and
W
2
W_2
W
2
are subspaces of a vector space
V
V
V
such that
V
=
W
1
⊕
W
2
V=W_1\oplus W_2
V
=
W
1
⊕
W
2
. Under what conditions we have
W
=
(
W
∩
W
1
)
⊕
(
W
∩
W
2
)
W=(W\cap W_1)\oplus(W\cap W_2)
W
=
(
W
∩
W
1
)
⊕
(
W
∩
W
2
)
?