SCHWEITZER 1978

For a distributive lattice $L$ , consider the following two statements:

(A) Every ideal of $L$ is the kernel of at least two different homomorphisms.

(B) $L$ contains no maximal ideal.

Which one of these statements implies the other?

(Every homomorphism $\varphi$ of $L$ induces an equivalence relation on $L$ : $a \sim b$ if and only if $a \varphi=b \varphi$ . We do not consider two homomorphisms different if they imply the same equivalence relation.)

J. Varlet, E. Fried

Suppose that the function $g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $g$ must be analytic. Is the statement also true for the interval $(-1,0)$ instead of $(0,1)$ ?

J. Kalina, L. Lempert

Let $T$ be a surjective mapping of the hyperbolic plane onto itself which maps collinear points into collinear points. Prove that $T$ must be an isometry.

M. Bognar

Let $X_1, \ldots ,X_n$ be $n$ points in the unit square ( $n>1$ ). Let $r_i$ be the distance of $X_i$ from the nearest point (other than $X_i$ ). Prove that the inequality $$r_1^2+ \ldots +r_n^2 \leq 4.$$

L. Fejes-Toth, E. Szemeredi

Let $Y_n$ be a binomial random variable with parameters $n$ and $p$ . Assume that a certain set $H$ of positive integers has a density and that this density is equal to $d$ . Prove the following statements:

(a) $\lim _{n \rightarrow \infty}P(Y_n\in H)=d$ if $H$ is an arithmetic progression.

(b) The previous limit relation is not valid for arbitrary $H$ .

(c) If $H$ is such that $P(Y_n \in H)$ is convergent, then the limit must be equal to $d$ .

L. Posa