CZ-SK-MO 1986

Given $n \in N$ , let $A$ be a family of subsets of $\{1,2,...,n\}$ . If for every two sets $B,C \in A$ the set $(B \cup C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .

Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$ . If $x_1,...,x_m$ ( $n\ge m\ge3$ ) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$ , prove that $P$ cannot have integer roots$.

Prove that the entire space can be partitioned into "crosses" made of seven unit cubes as shown in the picture.

Let $C_1,C_2$ , and $C_3$ be points inside a bounded convex planar set $M$ . Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$ . Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$ , then $A\cap B \ne \emptyset$

A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$ . Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$ .

Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$ . Prove that the set $M$ contains no more than $9$ elements. Decide whether there exists such set M.