8th Austrian-Polish 1985


1. a, b, c are distinct non-zero reals with sum zero. Let x = (b-c)/a, y = (c-a)/b, z = (a-b)/c. Show that (x + y + z)(1/x + 1/y + 1/z) = 9.
2. For which n > 7 is there a graph with n points, such that there are 3 points of degree n-1, 3 points of degree n-2 and one person of degree k for k = 4, 5, 6 ... , n-3?
3. Four points form a convex quadrilateral with area 1, show that the sum of the six distances between each pair of points is at least 4 + √8.
4. Find all real solutions to: x⁴ + y² - xy³ = 9x/8 y⁴ + x² - x³y = 9y/8.
5. We have N identical sets of weights. Each set has four weights, each a different natural number. There is a subset of the 4N weights which weighs k for k = 1, 2, 3, ... , 1985. The weights and N are chosen so that the total weight of the 4N weights is as small as possible. How many such minimal sets are there?
6. ABCD is a tetrahedron. P is a point inside. The centroids of PBCD, APCD, ABPD, ABCP are A', B', C', D' respectively. Show that the volume of A'B'C'D' is 1/64 the volume of ABCD.
7. Find the least upper bound for the set of values (x₁x₂ + 2x₂x₃ + x₃x₄)/(x₁² + x₂² + x₃² + x₄²), where x_i are reals, not all zero.
8. A convex n-gon A₀A₁ ... A_n is divided into n-2 triangles by diagonals which do not intersect (except possibly at vertices of the n-gon). Find the number of ways of labeling the triangles T₁, T₂, ... , T_n-2, so that A_i is a vertex of T_i for each i. The diagram shows a possible labeling.
9. Given any convex polygon, show that we can find a point P inside the polygon and three vertices X, Y, Z, such that each of the two angles between PX and a side at X is acute, and similarly for Y and Z. The diagram below shows a poor choice of P. There is only vertex X satisfying the condition. Asterisks mark obtuse angles.

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