AUSTRIAN-POLISH 1988

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.

If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ( $n \ge 2$ ), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$ -tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$ .

Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$ .

Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$ . Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$ , the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?

We are given $1988$ unit cubes. Using some or all of these cubes, we form three quadratic boards $A, B,C$ of dimensions $a \times a \times 1$ , $b \times b \times 1$ , and $c \times c \times 1$ respectively, where $a \le b \le c$ . Now we place board $B$ on board $C$ so that each cube of $B$ is precisely above a cube of $C$ and $B$ does not overlap $C$ . Similarly, we place $A$ on $B$ . This gives us a three-floor tower. What choice of $a, b$ and $c$ gives the maximum number of such three-floor towers?

For a rectangle $R$ with integral side lengths, denote by $D(a, b)$ the number of ways of covering $R$ by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of $R$ . Determine the perimeter $P$ of the rectangle $R$ for which $\frac{D(a,b)}{a+b}$ is maximal.