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Miklós Schweitzer
1958 Miklós Schweitzer
1958 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklós Schweitzer 1958- Problem 11
11. Let
a
n
=
(
−
1
)
n
(
n
=
1
,
2
,
…
,
2
N
)
a_n = (-1)^n (n= 1, 2, \dots , 2N)
a
n
=
(
−
1
)
n
(
n
=
1
,
2
,
…
,
2
N
)
. Denote by
A
N
(
x
)
A_{N}(x)
A
N
(
x
)
the number of the sequences
1
≤
i
1
<
i
2
<
⋯
<
i
N
≤
2
N
1 \leq i_1 < i_2< \dots <i_N \leq 2N
1
≤
i
1
<
i
2
<
⋯
<
i
N
≤
2
N
such that
a
i
1
+
a
i
2
+
⋯
+
a
i
N
<
x
N
2
(
−
∞
<
x
<
∞
)
a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)
a
i
1
+
a
i
2
+
⋯
+
a
i
N
<
x
2
N
(
−
∞
<
x
<
∞
)
. Show that
lim
N
→
∞
A
N
(
x
)
(
2
N
N
)
=
1
2
π
∫
−
∞
∞
e
−
u
2
2
d
u
\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du
lim
N
→
∞
(
N
2
N
)
A
N
(
x
)
=
2
π
1
∫
−
∞
∞
e
−
2
u
2
d
u
.(N. 16)
10
1
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Miklós Schweitzer 1958- Problem 10
10. Prove that the function
f
(
x
)
=
∫
−
∞
∞
(
sin
θ
θ
)
2
k
cos
(
2
x
θ
)
d
θ
f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta
f
(
x
)
=
∫
−
∞
∞
(
θ
s
i
n
θ
)
2
k
cos
(
2
x
θ
)
d
θ
where
k
k
k
is a positive integer, satisfies the following conditions:(i)
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
if
∣
x
∣
≥
k
\mid x \mid \geq k
∣
x
∣≥
k
and
f
(
x
)
≥
0
f(x) \geq 0
f
(
x
)
≥
0
elsewhere; (ii) in interval
(
l
,
l
+
1
)
(l,l+1)
(
l
,
l
+
1
)
(
l
=
−
k
,
−
k
+
1
,
…
,
k
−
1
)
(l= -k, -k+1, \dots , k-1)
(
l
=
−
k
,
−
k
+
1
,
…
,
k
−
1
)
the function
f
(
x
)
f(x)
f
(
x
)
is a polynomial of degree
2
k
−
1
2k-1
2
k
−
1
at most. (R. 7)
9
1
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Miklós Schweitzer 1958- Problem 9
9. Show that if
f
(
z
)
=
1
+
a
1
z
+
a
2
z
2
+
…
f(z) = 1+a_1 z+a_2z^2+\dots
f
(
z
)
=
1
+
a
1
z
+
a
2
z
2
+
…
for
∣
z
∣
≤
1
\mid z \mid\leq 1
∣
z
∣≤
1
and
1
2
π
∫
0
2
π
∣
f
(
e
i
ϕ
)
∣
2
d
ϕ
<
(
1
+
∣
a
1
∣
2
4
)
2
\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2
2
π
1
∫
0
2
π
∣
f
(
e
i
ϕ
)
∣
2
d
ϕ
<
(
1
+
4
∣
a
1
∣
2
)
2
,then
f
(
z
)
f(z)
f
(
z
)
has a root in the disc
∣
z
∣
≤
1
\mid z \mid \leq 1
∣
z
∣≤
1
.(F. 4)
8
1
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Miklós Schweitzer 1958- Problem 8
8. Let the function
f
(
x
)
f(x)
f
(
x
)
be periodic with the period
1
1
1
, non-negative, concave in the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
and continuous at the point
0
0
0
. Prove that
f
(
n
x
)
≤
n
f
(
x
)
f(nx)\leq nf(x)
f
(
n
x
)
≤
n
f
(
x
)
for every real
x
x
x
and positive integer
n
n
n
. (R. 6)
7
1
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Miklós Schweitzer 1958- Problem 7
7. Let
a
0
a_0
a
0
and
a
1
a_1
a
1
be arbitrary real numbers, and let
a
n
+
1
=
a
n
+
2
n
+
1
a
n
−
1
a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}
a
n
+
1
=
a
n
+
n
+
1
2
a
n
−
1
(
n
=
1
,
2
,
…
)
(n= 1, 2, \dots)
(
n
=
1
,
2
,
…
)
Show that the sequence
(
a
n
n
2
)
n
=
1
∞
\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}
(
n
2
a
n
)
n
=
1
∞
is convergent and find its limit. (S. 10)
6
1
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Miklós Schweitzer 1958- Problem 6
6. Prove that if
a
n
≥
0
a_n \geq 0
a
n
≥
0
and
1
n
∑
k
=
1
n
a
k
≥
∑
k
=
n
+
1
2
n
a
k
\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k
n
1
∑
k
=
1
n
a
k
≥
∑
k
=
n
+
1
2
n
a
k
(
n
=
1
,
2
,
…
)
(n=1, 2, \dots)
(
n
=
1
,
2
,
…
)
,then
∑
k
=
1
∞
a
k
\sum_{k=1}^{\infty} a_k
∑
k
=
1
∞
a
k
is convergent and its sum is less than
2
e
a
1
2ea_1
2
e
a
1
. (S. 9)
5
1
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Miklós Schweitzer 1958- Problem 5
5. Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. (St. 2)
4
1
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Miklós Schweitzer 1958- Problem 4
4. Let
P
1
P
2
P
3
P
4
P
5
P
6
P_1 P_2 P_3 P_4 P_5 P_6
P
1
P
2
P
3
P
4
P
5
P
6
be a convex hexagon. Denote by
T
T
T
its area and by
t
t
t
the area of the triangle
Q
1
Q
2
Q
3
Q_1 Q_2 Q_3
Q
1
Q
2
Q
3
, where
Q
1
,
Q
2
Q_1,Q_2
Q
1
,
Q
2
and
Q
3
Q_3
Q
3
are the midpoints of
P
1
P
4
,
P
2
P
5
,
P
3
P
6
P_1P_4,P_2P_5,P_3P_6
P
1
P
4
,
P
2
P
5
,
P
3
P
6
respectively. Prove that
t
<
1
4
T
t<\frac{1}{4}T
t
<
4
1
T
. (G. 3)
3
1
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Miklós Schweitzer 1958- Problem 3
3. Let
n
n
n
be a positive integer having at least one prime factor with expoente
≥
2
\geq 2
≥
2
. Show that
n
n
n
has as many factorizations into an odd number of factors as into an even number of factors. (Factorizations into the same factors arranged in different order are considered different.)(N. 10)
2
1
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Miklós Schweitzer 1958- Problem 2
2. Let
A
(
x
)
A(x)
A
(
x
)
denote the number of positive integers
n
n
n
not greater than
x
x
x
and having at least one prime divisor greater than
n
3
\sqrt[3]{n}
3
n
. Prove that
lim
x
→
∞
A
(
x
)
x
\lim_{x\to \infty} \frac {A(x)}{x}
lim
x
→
∞
x
A
(
x
)
exists. (N. 15)
1
1
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Miklós Schweitzer 1958- Problem 1
1. Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) (A. 14)