SCHWEITZER 1995

Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.

Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$ , prove that there exists an interval I st $\int_I f = \int_I g=\frac12$ .

Denote $\langle x\rangle$ the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers $a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n$ such that $$f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle$$ for every x iff there is a k such that $$\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)$$ is constant.

For odd numbers $a_1 , ..., a_k$ and even numbers $b_1 , ..., b_k$ , we know that $\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n$ is satisfied for n = 1,2, ..., N. Prove that $k\geq 2^N$ and that for $k = 2^N$ there exists a solution $(a_1,...,b_1,...)$ with the above properties.

Let A be a subset of the set $\{1,2, ...,n\}$ with at least $100\sqrt n$ elements. Prove that there is a four-element arithmetic sequence in which each element is the sum of two different elements of the set A.

Prove that every finite triangle-free graph can be embedded as an induced subgraph in a finite triangle-free graph of diameter 2.

Let P be a finite, partially ordered set with one largest element, which is the only upper bound of the set of minimal elements. Prove that any monotonic function $f : P^n\to P$ can be written in the form $g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m )$ , where $c_i\in P$ and g is a monotonic, idempotent function. (g is idempotent iff $g(x , x , ..., x) = x\,\forall x\in P$ )

A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ( $P_m = P_i$ for some i < m).

Let $X =\{ X_1 , X_2 , ...\}$ be a countable set of points in space. Show that there is a positive sequence $\{a_k\}$ such that for any point $Z\not\in X$ the distance between the point Z and the set $\{X_1,X_2 , ...,X_k\}$ is at least $a_k$ for infinitely many k.

Let F(x) be a known distribution function, the random variables $\eta_1 , \eta_2 ...$ be independent of the common distribution function $F( x - \vartheta)$ , where $\vartheta$ is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of $\varepsilon> 0$ there exist a Lebesgue measure $\varepsilon$ Borel set E ("confidence set") and a Borel-measurable function $t_n( x_1 ,. .., x_n )$ ( n = 1,2, ...) such that for any $\vartheta$ we have $$P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} ( n > n_0 ( \varepsilon, F ) )$$ Prove that a) if F is not absolutely continuous, then the shift parameter is "well estimated", b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".