SCHWEITZER 1992

Let p be a prime and $a_1 , a_2 , ..., a_k$ pairwise incongruent modulo p . Prove that $[\sqrt {k-1}]$ of the elements can be selected from $a_i$ 's such that adding any numbers different from the selected ones will never give a number divisible by p .

Call a (non-trivial) lattice class a pseudo-variety if it is closed under taking a homomorphic image, a direct product, and a convex subset. Prove that the smallest distributive pseudo-variety cannot be defined by a first-order set of formulas.

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$ , whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$ , there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

Prove that if the $a_i$ 's are different natural numbers, then $\sum_ {j = 1}^n a_j ^ 2 \prod_{k \neq j} \frac{a_j + a_k}{a_j-a_k}$ is a square number.

Let $E \subset [0,1]$ be a Lebesgue measurable set having Lebesgue measure $| E |<\frac{1}{2}$ . Let $$h (s) = \int _ {\overline {E}} \frac{dt}{{(s-t)}^2}$$ where $\overline {E} = [0,1] \backslash E$ . Prove that there is one $t \in \overline {E}$ for which $$\int_E \frac {ds} {h (s) {(s-t)} ^ 2} \leq c {| E |} ^ 2$$ with some absolute constant c .

Prove that in a topological space X , if all discrete subspaces have compact closure , then X is compact.

Let $F$ be a set of filters on X so that if $\sigma, \tau \in F$ , $\forall S \in\sigma$ , $\forall T\in\tau$ , we have $S \cap T\neq\emptyset$ , then $\sigma \cap \tau \in F$ . We say that $F$ is compatible with a topology on X when $x \in X$ is a contact point of $A\subset X$ , if and only if , there is $\sigma \in F$ such that $x \in S$ and $S \cap A \neq\emptyset$ for all $S \in\sigma$ .

When is there an $F$ compatible with the topology on X in which finite subsets of X and X are closed ?

contact point is also known as adherent point.

Let K be a bounded, d-dimensional convex polyhedron that is not simplex and P is a point on K. Show that if vertices $P_1 , ..., P_k$ are not all on the same face of K, then one of them can be omitted so that the convex hull of the remaining vertices of K still contains P.

We place n points in the unit square independently, according to a uniform distribution. These points are the vertices of a graph $G_n$ . Two points are connected by an edge if the slope of the segment connecting them is nonnegative. Denote by $M_n$ the event that the graph $G_n$ has a 1-factor. Prove that $\lim_{n \to \infty} P(M_ {2n}) = 1$ .