11th Austrian-Polish 1988


1. p(x) is a polynomial with integer coefficients and at least 6 distinct integer roots. Show that p(x) - 12 has no integer roots.
2. 1 ≤ a₁ ≤ a₂ ≤ ... ≤ a_n are positive integers. If a₂ ≥ 2, show that (a₁x₁² + ... + a_nx_n²) + 2(x₁x₂ + x₂x₃ + ... + x_n-1x_n) > 0 for all real x_i which are not all zero. If a₂ < 2, show that we can find x_i not all zero for which the inequality is false.
3. ABCD is a convex quadrilateral with no two sides parallel. The bisectors of the angles formed by the two pairs of opposite sides meet the sides of ABCD at the points P, Q, R, S, so that PQRS is convex. Show that ABCD has a circumcircle iff PQRS is a rhombus.
4. Find all strictly increasing real-valued functions on the reals such that f( f(x) + y) = f(x + y) + f(0) for all x, y.
5. The integer sequences a₁, a₂, a₃, ... and b₁, b₂, b₃, ... satisfy b_n = a_n + 9, a_n+1 = 8b_n + 8. The number 1988 appears in at least one of the sequences. Show that the sequence a_n does not contain a square.
6. R₁, R₂, R₃ are three rays (in space) with endpoint O such that if A_i is any point on R_i except O, then the triangle A₁A₂A₃ is acute-angled. Show that the three rays are mutually perpendicular.
7. 8 points are arranged in a circle. Each point is colored yellow or blue. Once a minute the colors are all changed simultaneously. If the two neighbors of a point were the same color, then the point becomes yellow. If the two neighbors are opposite colors, then the point becomes blue. Show that whatever the initial colors, all points eventually become yellow. What is the maximum time required to achieve this state?
8. Using at most 1988 unit cubes form three boards: A measuring 1 x a x a, B 1 x b x b and C 1 x c x c, where a ≤ b ≤ c. Let d(a, b, c) be the number of ways of placing B flat on C and A flat on B, so that the small cubes line up, the footprint of B lies within C (so it does not overhang), and similarly the footprint of A lies within B. We regard arrangements as distinct even if one can be rotated or reflected into the other. Find a, b, c which maximize d(a, b, c).
9. The rectangle R has integral sides a and b. Let D(a, b) be the number of ways of tiling it with congruent rectangular tiles which are all similarly oriented. Find the value of a+b which maximizes D(a, b)/(a+b).

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