1962 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

(10)

1

Miklos Schweitzer 1962_10

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

1

Miklos Schweitzer 1962_9

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

1

Miklos Schweitzer 1962_8

M(r,f)

the maximum modulus on the circle |z|\equal{}r of the transcendent entire function

f(z)

M_n(r,f)

nth

partial sum of the power series of

f(z)

. Prove that the existence of an entire function

f_0(z)

and a corresponding sequence of positive numbers r_1

1

Miklos Schweitzer 1962_7

Prove that the function

f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}

(where the positive value of the square root is taken) is monotonically decreasing in the interval

0<\nu<1

1

Miklos Schweitzer 1962_6

E

be a bounded subset of the real line, and let

\Omega

be a system of (non degenerate) closed intervals such that for each

x \in E

there exists an

I \in \Omega

with left endpoint

x

. Show that for every

\varepsilon > 0

there exists a finite number of pairwise non overlapping intervals belonging to

\Omega

E

with the exception of a subset of outer measure less than

\varepsilon

. [J. Czipszer]

1

Miklos Schweitzer 1962_5

f

be a finite real function of one variable. Let

\overline{D}f

\underline{D}f

be its upper and lower derivatives, respectively, that is, \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k} , \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}. Show that

\overline{D}f

\underline{D}f

are Borel-measurable functions. [A. Csaszar]

1

Miklos Schweitzer 1962_4

Show that \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(mod\;p\ ) for every prime

p\equiv 3 \;(<span class='latex-bold'>mod</span>\;4\ )

1

Miklos Schweitzer 1962_3

A

B

be two Abelian groups, and define the sum of two homomorphisms

\eta

\chi

A

B

by a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ . With this addition, the set of homomorphisms from

A

B

forms an Abelian group

H

. Suppose now that

A

p

p

a prime number). Prove that in this case

H

becomes a topological group under the topology defined by taking the subgroups p^kH \;(k\equal{}1,2,...) as a neighborhood base of

0

H

is complete in this topology and that every connected component of

H

consists of a single element. When is

H

compact in this topology? [L. Fuchs]

1

Miklos Schweitzer 1962_2

Determine the roots of unity in the field of

p

-adic numbers. L. Fuchs

1

Miklos Schweitzer 1962_1

f

g

be polynomials with rational coefficients, and let

F

G

denote the sets of values of

f

g

at rational numbers. Prove that F \equal{} G holds if and only if f(x) \equal{} g(ax \plus{} b) for some suitable rational numbers a\not \equal{} 0 and

b