1966 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

(10)

1

Miklos Schweitzer 1966_10

For a real number

x

in the interval

(0,1)

with decimal representation

0.a_1(x)a_2(x)...a_n(x)...,

n(x)

the smallest nonnegative integer such that \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 . Determine

\int_0^1n(x)dx

\overline{abcd}

denotes the decimal number with digits

a,b,c,d .

1

Miklos Schweitzer 1966_9

\sum_{m=-\infty}^{+\infty} |a_m| < \infty

, then what can be said about the following expression?

\lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.

1

Miklos Schweitzer 1966_8

Prove that in Euclidean ring

R

the quotient and remainder are always uniquely determined if and only if

R

is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases:

R

can also be a field or the null ring.) E. Fried

1

Miklos Schweitzer 1966_7

Does there exist a function

f(x,y)

of two real variables that takes natural numbers as its values and for which f(x,y)\equal{}f(y,z) implies x\equal{}y\equal{}z? A. Hajnal

1

Miklos Schweitzer 1966_6

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between \minus{}3 degrees and \plus{}5 degrees." Show that it would suffice to say, "Both \minus{}3 degrees and \plus{}5 degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) A.Csaszar

1

Miklos Schweitzer 1966_5

T

" erected at point

A

x

xy

-plane is the union of a segment

AB

in the upper half-plane perpendicular to the

x

-axis and a segment

CD

B

in its interior and parallel to the

x

-axis. Show that it is impossible to erect a letter

T

at every point of the

x

-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points. A.Csaszar

1

Miklos Schweitzer 1966_4

I

be an ideal of the ring

\mathbb{Z}\left[x\right]

of all polynomials with integer coefficients such that
a) the elements of

I

do not have a common divisor of degree greater than

0

I

contains of a polynomial with constant term

1

I

contains the polynomial

1 + x + x^2 + ... + x^{r-1}

for some natural number

r

1

Miklos Schweitzer 1966_3

f(n)

denote the maximum possible number of right triangles determined by

n

coplanar points. Show that \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 . P. Erdos

1

Miklos Schweitzer 1966_2

Characterize those configurations of

n

coplanar straight lines for which the sum of angles between all pairs of lines is maximum. L.Fejes-Toth, A. Heppes

1

Miklos Schweitzer 1966_1

Show that a segment of length

h

can go through or be tangent to at most 2\lfloor h/\sqrt{2}\rfloor\plus{}2 nonoverlapping unit spheres. L.Fejes-Toth, A. Heppes