20th Austrian-Polish 1997


1. Four circles, none of which lies inside another, pass through the point P. Two circles touch the line L at P and the other two touch the line M at P. The other points of intersection of the circles are A, B, C, D. Show that A, B, C, D lie on a circle iff L and M are perpendicular.
2. A piece is on each square of an m x n board. The allowed move for each piece is h squares parallel to the bottom edge of the board and k squares parallel to the sides. How many ways can we move every piece simultaneously so that after the move there is still one piece on each square?
3. The 97 numbers 49/1, 49/2, 49/3, ... , 49/97 are written on a blackboard. We repeatedly pick two numbers a, b on the board and replace them by 2ab - a - b + 1 until only one number remains. What are the possible values of the final number?
4. ABCD is a convex quadrilateral with AB parallel to CD. The diagonals meet at E. X is the midpoint of the line joining the orthocenters of BEC and AED. Show that X lies on the perpendicular to AB through E.
5. Show that no cubic with integer coefficients can take the value ±3 at each of four distinct primes.
6. Show that there is no integer-valued function on the integers such that f(m + f(n) ) = f(m) - n for all m, n.
7. Show that x² + y² + 1 > x(y + 1) for all reals x, y. Find the largest k such that x² + y² + 1 ≥ kx(y + 1) for all reals x, y. Find the largest k such that m² + n² + 1 ≥ km(n + 1) for all integers m, n.
8. Let X be a set with n members. Find the largest number of subsets of X each with 3 members so that no two are disjoint.
9. k > 0 and P is a solid parallelepiped. S is the set of all points X for which there is a point Y in P such that XY ≤ k. Show that the volume of S = V + Fk + πEk²/4 + 4πk³/3, where V, F, E are respectively the volume, surface area and total edge length of P.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.