NO-ABEL 2001

Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$ . (For example, $A = \{1,2,3\}$ , then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$ .) A subset $F$ of P $(A)$ is called strong if the following is true: If $B_1$ and $B_2$ are elements of $F$ , then $B_1 \cup B_2$ is also an element of $F$ . Suppose that $F$ and $G$ are strong subsets of $P (A)$ . a) Is the union $F \cup G$ necessarily strong? b) Is the intersection $F \cap G$ necessarily strong?

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties