SCHWEITZER 1993

There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in $[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]$ for some $t_1, t_2$ , is at most $[\frac{n^2}{3}]$ , and the bound is sharp.

Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and $$\sum_{n\in C} \frac{1}{n} <\infty$$

Let K be the field formed by the addition of a root of the polynomial $x^4 - 2x^2 - 1$ to the rational field. Prove that there are no exceptional units in the ring of integers of K. (A unit $\varepsilon$ is called exceptional if $1-\varepsilon$ is also a unit.)

Let f be a ternary operation on a set of at least four elements for which (1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$ (2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$ for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ( $1 \leq i \leq n$ ) )

Does the set of real numbers have a well-order $\prec$ such that the intersection of the subset $\{(x,y) : x\prec y\}$ of the plane with every line is Lebesgue measurable on the line?

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$ . Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

Let H be a Hilbert space over the field of real numbers $\Bbb R$ . Find all $f: H \to \Bbb R$ continuous functions for which $$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$ $$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$ is satisfied for any $x , y , z \in H$ .

Let H be a complex separable Hilbert space and denote $B(H)$ the algebra of bounded linear operators on H. Find all *-subalgebras C of $B(H)$ for which for all $A \in B(H)$ and $T \in C$ there exists $S \in C$ that $$TA-AT^{\ast} = TS-ST^{\ast}$$ note: *-algebra is also known as involutive algebra.

Let $U_1 , U_2 , U_3$ be iid random variables on [0,1], which in order of magnitude, $U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}$ . Let $\alpha, p_1 , p_2 , p_3 \in [0,1]$ such that $P(U_j ^ {\ast} \ge p_j)= \alpha$ ( j = 1,2,3). Prove that $$P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha$$