AUSTRIAN-POLISH 1997

Each square of an $n \times m$ board is assigned a pair of coordinates $(x,y)$ with $1 \le x \le m$ and $1 \le y \le n$ . Let $p$ and $q$ be positive integers. A pawn can be moved from the square $(x,y)$ to $(x',y')$ if and only if $|x - x'| = p$ and $|y- y'| = q$ . There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done?

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$ .

Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$ .

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$ , . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$ .

Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$ , each with $3$ elements, so that no two of them are disjoint.

Given a parallelepiped $P$ , let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$ , let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$ . Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$ .