SCHWEITZER 1975

Show that there exists a tournament $(T,\rightarrow)$ of cardinality $\aleph_1$ containing no transitive subtournament of size $\aleph_1$ . ( A structure $(T,\rightarrow)$ is a $\textit{tournament}$ if $\rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $(T,\rightarrow)$ is transitive if $\rightarrow$ is transitive, that is, if it orders $T$ .)

A. Hajnal

Let $S$ be a semigroup without proper two-sided ideals and suppose that for every $a,b \in S$ at least one of the products $ab$ and $ba$ is equal to one of the elements $a,b$ . Prove that either $ab=a$ for all $a,b \in S$ or $ab=b$ for all $a,b \in S$ .

L. Megyesi

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $f \circ g$ defined by $${ (f \circ g)(n)= \sum_{d|n} f(d)g(\frac nd}).$$ (We call a complex number $\textit{complex rational}$ , if its real and imaginary parts are both rational.)

B. Csakany

Let $\{ f_n \}$ be a sequence of Lebesgue-integrable functions on $[0,1]$ such that for any Lebesgue-measurable subset $E$ of $[0,1]$ the sequence $\int_E f_n$ is convergent. Assume also that $\lim_n f_n=f$ exists almost everywhere. Prove that $f$ is integrable and $\int_E f=\lim_n \int_E f_n$ . Is the assertion also true if $E$ runs only over intervals but we also assume $f_n \geq 0 ?$ What happens if $[0,1]$ is replaced by $[0,+\infty) ?$

J. Szucs

Let $f$ be a differentiable real function and let $M$ be a positive real number. Prove that if $$|f(x+t)-2f(x)+f(x-t)| \leq Mt^2 \textrm{for all}\ x\ \textrm{and}\ t\ ,$$ then $$|f'(x+t)-f'(x)| \leq M|t|.$$

J. Szabados

Let $a<a'<b<b'$ be real numbers and let the real function $f$ be continuous on the interval $[a,b']$ and differentiable in its interior. Prove that there exist $c \in (a,b), c'\in (a',b')$ such that $$f(b)-f(a)=f'(c)(b-a),$$ $$f(b')-f(a')=f'(c')(b'-a'),$$ and $c<c'$ .

B. Szokefalvi Nagy

Prove that if $$\sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)$$ holds for a sequence $\{a_n \}$ of nonnegative real numbers with some positive integer $N$ , then $\alpha_{i+p} \geq p \alpha_i$ for $i,p=1,2,...,$ where $$\alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .$$

L. Leindler

Let $l_0,c,\alpha,g$ be positive constants, and let $x(t)$ be the solution of the differential equation $$([l_0+ct^{\alpha}] ^2x')'+g[l_0+ct^{\alpha}] \sin x=0, t \geq 0,\ \;-\frac{\pi}{2} <x< \frac{\pi}{2},$$ satisfying the initial conditions $x(t_0)=x_0, x'(t_0)=0$ . (This is the equation of the mathematical pendulum whose length changes according to the law $l=l_0+ct^{\alpha}$ .) Prove that $x(t)$ is defined on the interval $[t_0,\infty)$ ; furthermore, if $\alpha >2$ then for every $x_0 \not= 0$ there exists a $t_0$ such that $$\liminf_{t \rightarrow \infty} |x(t)| >0.$$

L. Hatvani

Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $0$ or $1$ .

J. Szucs

Let $X_1,X_2,...,X_n$ be (not necessary independent) discrete random variables. Prove that there exist at least $n^2/2$ pairs $(i,j)$ such that $$H(X_i+X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \},$$ where $H(X)$ denotes the Shannon entropy of $X$ .

GY. Katona

Assume that a face of a convex polyhedron $P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $P$ and passes through all vertices.

L .Lovasz