MathDB

1

Miklos Schweitzer 1983_12

X_1,X_2,\ldots, X_n

be independent, identically distributed, nonnegative random variables with a common continuous distribution function

F

. Suppose in addition that the inverse of

F

, the quantile function

Q

, is also continuous and

Q(0)=0

. Let

0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}

be the ordered sample from the above random variables. Prove that if

EX_1

is finite, then the random variable

\Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|

tends to zero with probability one as

n \rightarrow \infty

. S. Csorgp, L. Horvath

11

1

Miklos Schweitzer 1983_11

Let M^n \subset \mathbb{R}^{n\plus{}1} be a complete, connected hypersurface embedded into the Euclidean space. Show that

M^n

as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is,

M^n

can be decomposed to a direct product of the form M^n\equal{}M^k \times \mathbb{R}^{n\minus{}k} \;(k

M^k

is a hypersurface in some (k\plus{}1)-dimensional subspace E^{k\plus{}1} \subset \mathbb{R}^{n\plus{}1} , \mathbb{R}^{n\minus{}k} is the orthogonal complement of E^{k\plus{}1}. Z. Szabo

10

1

Miklos Schweitzer 1983_10

Let

R

be a bounded domain of area

t

in the plane, and let

C

be its center of gravity. Denoting by

T_{AB}

the circle drawn with the diameter

AB

, let

K

be a circle that contains each of the circles

T_{AB} \;(A,B \in R)

. Is it true in general that

K

contains the circle of area

2t

centered at

C

? J. Szucs

9

1

Miklos Schweitzer 1983_9

Prove that if

E \subset \mathbb{R}

is a bounded set of positive Lebesgue measure, then for every

u < 1/2

, a point x\equal{}x(u) can be found so that |(x\minus{}h,x\plus{}h) \cap E| \geq uh and |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh for all sufficiently small positive values of

h

. K. I. Koljada

8

1

Miklos Schweitzer 1983_8

Prove that any identity that holds for every finite

n

-distributive lattice also holds for the lattice of all convex subsets of the (n\minus{}1)-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice

n

-

<span class='latex-italic'>distributive</span>

if x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i)) holds for all elements of the lattice.) A. Huhn

7

1

Miklos Schweitzer 1983_7

Prove that if the function

f : \mathbb{R}^2 \rightarrow [0,1]

is continuous and its average on every circle of radius

1

equals the function value at the center of the circle, then

f

is constant. V. Totik

6

1

Miklos Schweitzer 1983_6

Let

T

be a bounded linear operator on a Hilbert space

H

, and assume that

\|T^n \| \leq 1

for some natural number

n

. Prove the existence of an invertible linear operator

A

on

H

such that \| ATA^{\minus{}1} \| \leq 1. E. Druszt

5

1

Miklos Schweitzer 1983_5

Let

g : \mathbb{R} \rightarrow \mathbb{R}

be a continuous function such that

x+g(x)

is strictly monotone (increasing or decreasing), and let

u : [0,\infty) \rightarrow \mathbb{R}

be a bounded and continuous function such that

u(t)+ \int_{t-1}^tg(u(s))ds

is constant on

[1,\infty)

. Prove that the limit

\lim_{t\rightarrow \infty} u(t)

exists. T. Krisztin

4

1

Miklos Schweitzer 1983_4

For which cardinalities

\kappa

do antimetric spaces of cardinality

\kappa

exist?

(X,\varrho)

is called an

<span class='latex-italic'>antimetric space</span>

if

X

is a nonempty set,

\varrho : X^2 \rightarrow [0,\infty)

is a symmetric map, \varrho(x,y)\equal{}0 holds iff x\equal{}y, and for any three-element subset

\{a_1,a_2,a_3 \}

of

X

\varrho(a_{1f},a_{2f})\plus{}\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f}) holds for some permutation

f

of

\{1,2,3 \}

. V. Totik

3

1

Miklos Schweitzer 1983_3

Let

f : \mathbb{R} \rightarrow \mathbb{R}

be a twice differentiable,

2 \pi

-periodic even function. Prove that if f''(x)\plus{}f(x)\equal{}\frac{1}{f(x\plus{} 3 \pi /2 )} holds for every

x

, then

f

is

\pi /2

-periodic. Z. Szabo, J. Terjeki

2

1

Miklos Schweitzer 1983_2

Let

I

be an ideal of the ring

R

and

f

a nonidentity permutation of the set

\{ 1,2,\ldots, k \}

for some

k

. Suppose that for every 0 \not\equal{} a \in R, \;aI \not\equal{} 0 and Ia \not\equal{}0 hold; furthermore, for any elements

x_1,x_2,\ldots ,x_k \in I

, x_1x_2\ldots x_k\equal{}x_{1f}x_{2f}\ldots x_{kf} holds. Prove that

R

is commutative. R. Wiegandt

1

Miklos Schweitzer 1983_1

Given

n

points in a line so that any distance occurs at most twice, show that the number of distance occurring exactly once is at least

\lfloor n/2 \rfloor

. V. T. Sos, L. Szekely

1983 Miklós Schweitzer

Subcontests

Miklos Schweitzer 1983_12

Miklos Schweitzer 1983_11

Miklos Schweitzer 1983_10

Miklos Schweitzer 1983_9

Miklos Schweitzer 1983_8

Miklos Schweitzer 1983_7

Miklos Schweitzer 1983_6

Miklos Schweitzer 1983_5

Miklos Schweitzer 1983_4

Miklos Schweitzer 1983_3

Miklos Schweitzer 1983_2

Miklos Schweitzer 1983_1