29th Swedish 1989


1. Show that in base n²+1 the numbers n²(n²+2)² and n⁴(n²+2)² have the same digits but in opposite order.
2. Find all continuous functions f(x) such that f(x) + f(x²) = 2 for all real x.
3. For which positive integers n is n³ - 18n² + 115n - 391 a cube?
4. ABCD is a regular tetrahedron. Find a point P on the edge BD such that the sphere diameter AP touches the edge CD.
5. The positive reals x₁, x₂, x₃, x₄, x₅ satisfy x₁ < x₂ and x₂ ≤ each of x₃, x₄, x₅. Also a > 0. Show that 1/(x₁+x₃)^a + 1/(x₂+x₄)^a + 1/(x₂+x₅)^a < 1/(x₁+x₂)^a + 1/(x₂+x₃)^a + 1/(x₄+x₅)^a.
6. 4n points are arranged around a circle. The points are colored alternately yellow and blue. The yellow points are divided into pairs and each pair is joined by a yellow line segment. Similarly for the blue points. At most two segments meet at any point inside the circle. Show that there are at least n points of intersection between a yellow segment and a blue segment.

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