SCHWEITZER 1984

Let $k$ be an arbitrary cardinality. Show that there exists a tournament $T_k = (V_n , E_n)$ such that for any coloring $f: E_n \to k$ of the edge set $E_n$ , there are three different vertices $x_0 , x_1 , x_2 \in V_n$ such that $x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n$ and $\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2$ (A tounament is a directed graph such that for any vertices $x, y \in V_n, x \neq y$ exactly one of the relations $xy \in E_n$ holds.) (C.19) [A. Hajnal]

Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$ ). (M.16) [M. Laczkovich]

Let $a$ and $b$ be positive integers such that when dividing them by any prime $p$ , the remainder of $a$ is always less than or equal to the remainder of $b$ . Prove that $a=b$ . (N.16) [P. Erdos, P. P. Pálify]

Let $x_1 , x_2 , y_1 , y_2 , z_1 , z_2$ be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on $\{x_1 , x_2 , y_1, y_2 \}$ , $\{ x_1 , x_2 , z_1 , z_2 \}$ and $\{y_1 , y_2 , z_1 , z_2 \}$ are algebraically dependent. Prove that there exists a transcendental number $t$ that depends algebraically on each of the pairs $\{ x_1 , x_2\}$ , $\{ y_1 , y_2 \}$ , and $\{ z_1 , z_2 \}$ . (A.37) [L. Lovász]

Let $a_0 , a_1 , \dots$ be nonnegative real numbers such that $\sum_{n=0}^{\infty}a_n = \infty$ For arbitrary $c>0$ , let $n_{j}(c)= \min \left \{ k : c.j \leq \sum_{i=0}^{k} a_i \right \}$ , $j= 1,2, \dots$ Prove that if $\sum_{i=0}^{\infty}a_i^2 = \infty$ , then there exists a $c>0$ for which $\sum_{j=1}^{\infty} a_{n_j (c)} = \infty$ .(S.24) [P. Erdos, I. Joó, L. Székely]

For which Lebesgue-measurable subsets $E$ of the real line does a positive constant $c$ exist for which $\sup_{-\infty < t<\infty} \left | \int_{E} e^{itx} f(x) dx\right | \leq c \sup_{n=0,\pm 1,\dots } \left | \int_{E} e^{inx} f(x) dx\right |$ for all integrable functions $f$ on $E$ ? (M.17) [G. Halász]

Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$ , that is, for every $f \in V$ , $\int_{0}^{1} f(x)P(x)dx =0$ (F.39) [A. Pinkus, V. Totik]

Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? (G.36) [L. Fejes-Tóth]

Let $X_0, X_1, \dots$ be independent, indentically distributed, nondegenerate random variables, and let $0<\alpha <1$ be a real number. Assume that the series $\sum_{k=1}^{\infty} \alpha^{k} X_k$ is convergent with probability one. Prove that the distribution function of the sum is continuous. (P. 23) [T. F. Móri]

Let $X_1, X_2, \dots$ be independent random variables with the same distribution $P(X_i = 1) = P(X_i = -1)=\frac{1}{2} (i= 1, 2, \dots )$ Define $S_0=0, Sn=X_1 +X_2+\dots +X_n (n=1, 2, \dots$ ), $\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right | (x=0, \pm 1, \pm 2, \dots$ ),

and $\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right | (n=0,1,\dots$ ).

Prove that $P(\lim \inf \alpha(n)=0) =1$ and that there is a number $0<c<\infty$ such that $P(\lim \inf \alpha(n)/\log n=c) =1$ (P.24) [P. Révész]