SCHWEITZER 1966

Show that a segment of length $h$ can go through or be tangent to at most $2\lfloor h/\sqrt{2}\rfloor+2$ nonoverlapping unit spheres. L.Fejes-Toth, A. Heppes

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

Let $f(n)$ denote the maximum possible number of right triangles determined by $n$ coplanar points. Show that $$\lim_{n\rightarrow \infty} \frac{f(n)}{n^2}=\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}=0 .$$

P. Erdos

Let $I$ 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 $0$ , and

b) $I$ contains of a polynomial with constant term $1$ .

Prove that $I$ contains the polynomial $1 + x + x^2 + ... + x^{r-1}$ for some natural number $r$ .

Gy. Szekeres

A "letter $T$ " erected at point $A$ of the $x$ -axis in the $xy$ -plane is the union of a segment $AB$ in the upper half-plane perpendicular to the $x$ -axis and a segment $CD$ containing $B$ 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

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

A.Csaszar

Does there exist a function $f(x,y)$ of two real variables that takes natural numbers as its values and for which $f(x,y)=f(y,z)$ implies $x=y=z?$

A. Hajnal

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

If $\sum_{m=-\infty}^{+\infty} |a_m| < \infty$ , then what can be said about the following expression? $$\lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.$$

P. Turan

For a real number $x$ in the interval $(0,1)$ with decimal representation

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

denote by $n(x)$ the smallest nonnegative integer such that

$$\overline{a_{n(x)+1}a_{n(x)+2}a_{n(x)+3}a_{n(x)+4}}=1966 .$$

Determine $\int_0^1n(x)dx$ . ( $\overline{abcd}$ denotes the decimal number with digits $a,b,c,d .$ )

A. Renyi