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Problems
Contests
Undergraduate contests
IMS
2006 IMS
2006 IMS
Part of
IMS
Subcontests
(5)
1
1
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Sum
Prove that for each
m
≥
1
m\geq1
m
≥
1
:
∑
∣
k
∣
<
m
(
2
m
m
+
k
)
≥
2
2
m
−
1
\sum_{|k|<\sqrt m}\binom{2m}{m+k}\geq 2^{2m-1}
∣
k
∣
<
m
∑
(
m
+
k
2
m
)
≥
2
2
m
−
1
[hide="Hint"]Maybe probabilistic method works
3
1
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Finite order
G
G
G
is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of
G
G
G
which have finite order is a subgroup og
G
G
G
.
2
1
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Partition of Natural Numbers
For each subset
C
C
C
of
N
\mathbb N
N
, Suppose
C
⊕
C
=
{
x
+
y
∣
x
,
y
∈
C
,
x
≠
y
}
C\oplus C=\{x+y|x,y\in C, x\neq y\}
C
⊕
C
=
{
x
+
y
∣
x
,
y
∈
C
,
x
=
y
}
. Prove that there exist a unique partition of
N
\mathbb N
N
to sets
A
A
A
,
B
B
B
that
A
⊕
A
A\oplus A
A
⊕
A
and
B
⊕
B
B\oplus B
B
⊕
B
do not have any prime numbers.
4
1
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Countable
Assume that
X
X
X
is a seperable metric space. Prove that if
f
:
X
⟶
R
f: X\longrightarrow\mathbb R
f
:
X
⟶
R
is a function that
lim
x
→
a
f
(
x
)
\lim_{x\rightarrow a}f(x)
lim
x
→
a
f
(
x
)
exists for each
a
∈
R
a\in\mathbb R
a
∈
R
. Prove that set of points in which
f
f
f
is not continuous is countable.
5
1
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Complex number
Suppose that
a
1
,
a
2
,
…
,
a
k
∈
C
a_{1},a_{2},\dots,a_{k}\in\mathbb C
a
1
,
a
2
,
…
,
a
k
∈
C
that for each
1
≤
i
≤
k
1\leq i\leq k
1
≤
i
≤
k
we know that
∣
a
k
∣
=
1
|a_{k}|=1
∣
a
k
∣
=
1
. Suppose that
lim
n
→
∞
∑
i
=
1
k
a
i
n
=
c
.
\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.
n
→
∞
lim
i
=
1
∑
k
a
i
n
=
c
.
Prove that
c
=
k
c=k
c
=
k
and
a
i
=
1
a_{i}=1
a
i
=
1
for each
i
i
i
.