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Miklós Schweitzer
1964 Miklós Schweitzer
1964 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1964_10
Let
ε
1
,
ε
2
,
.
.
.
,
ε
2
n
\varepsilon_1,\varepsilon_2,...,\varepsilon_{2n}
ε
1
,
ε
2
,
...
,
ε
2
n
be independent random variables such that P(\varepsilon_i\equal{}1)\equal{}P(\varepsilon_i\equal{}\minus{}1)\equal{}\frac 12 for all
i
i
i
, and define S_k\equal{}\sum_{i\equal{}1}^k \varepsilon_i, \;1\leq k \leq 2n. Let
N
2
n
N_{2n}
N
2
n
denote the number of integers
k
∈
[
2
,
2
n
]
k\in [2,2n]
k
∈
[
2
,
2
n
]
such that either
S
k
>
0
S_k>0
S
k
>
0
, or S_k\equal{}0 and S_{k\minus{}1}>0. Compute the variance of
N
2
n
N_{2n}
N
2
n
.
9
1
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Miklos Schweitzer 1964_9
Let
E
E
E
be the set of all real functions on I\equal{}[0,1]. Prove that one cannot define a topology on
E
E
E
in which
f
n
→
f
f_n\rightarrow f
f
n
→
f
holds if and only if
f
n
f_n
f
n
converges to
f
f
f
almost everywhere.
8
1
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Miklos Schweitzer 1964_8
Let
F
F
F
be a closed set in the
n
n
n
-dimensional Euclidean space. Construct a function that is
0
0
0
on
F
F
F
, positive outside
F
F
F
, and whose partial derivatives all exist.
7
1
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Miklos Schweitzer 1964_7
Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution
f
(
t
)
f(t)
f
(
t
)
and any real number c,f(t\plus{}c) is also a solution.
6
1
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Miklos Schweitzer 1964_6
Let
y
1
(
x
)
y_1(x)
y
1
(
x
)
be an arbitrary, continuous, positive function on
[
0
,
A
]
[0,A]
[
0
,
A
]
, where
A
A
A
is an arbitrary positive number. Let
y
n
+
1
=
2
∫
0
x
y
n
(
t
)
d
t
(
n
=
1
,
2
,
.
.
.
)
.
y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .
y
n
+
1
=
2
∫
0
x
y
n
(
t
)
d
t
(
n
=
1
,
2
,
...
)
.
Prove that the functions
y
n
(
x
)
y_n(x)
y
n
(
x
)
converge to the function
y
=
x
2
y=x^2
y
=
x
2
uniformly on
[
0
,
A
]
[0,A]
[
0
,
A
]
.
5
1
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Miklos Schweitzer 1964_5
Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?
4
1
Hide problems
Miklos Schweitzer 1964_4
Let
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
be the vertices of a closed convex
n
n
n
-gon
K
K
K
numbered consecutively. Show that at least n\minus{}3 vertices
A
i
A_i
A
i
have the property that the reflection of
A
i
A_i
A
i
with respect to the midpoint of A_{i\minus{}1}A_{i\plus{}1} is contained in
K
K
K
. (Indices are meant
mod
n
.
\textrm{mod} \;n\ .
mod
n
.
)
3
1
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Miklos Schweitzer 1964_3
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.
2
1
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Miklos Schweitzer 1964_2
Let
p
p
p
be a prime and let l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ . be homogeneous linear polynomials with integral coefficients. Suppose that for every pair
(
ξ
,
η
)
(\xi,\eta)
(
ξ
,
η
)
of integers, not both divisible by
p
p
p
, the values
l
k
(
ξ
,
η
)
,
1
≤
k
≤
p
2
l_k(\xi,\eta), \;1\leq k\leq p^2
l
k
(
ξ
,
η
)
,
1
≤
k
≤
p
2
, represent every residue class
mod
p
\textrm{mod} \;p
mod
p
exactly
p
p
p
times. Prove that the set of pairs
{
(
a
k
,
b
k
)
:
1
≤
k
≤
p
2
}
\{(a_k,b_k): 1\leq k \leq p^2 \}
{(
a
k
,
b
k
)
:
1
≤
k
≤
p
2
}
is identical
mod
p
\textrm{mod} \;p
mod
p
with the set \{(m,n): 0\leq m,n \leq p\minus{}1 \}.
1
1
Hide problems
Miklos Schweitzer 1964_1
Among all possible representations of the positive integer
n
n
n
as n\equal{}\sum_{i\equal{}1}^k a_i with positive integers
k
,
a
1
<
a
2
<
.
.
.
<
a
k
k, a_1 < a_2 < ...<a_k
k
,
a
1
<
a
2
<
...
<
a
k
, when will the product \prod_{i\equal{}1}^k a_i be maximum?