26th Swedish 1986


1. Show that x⁶ - x⁵ + x⁴ - x³ + x² - x + 3/4 has no real zeros.
2. ABCD is a quadrilateral area S. Its diagonals meet at X. Area ABX = X₁, area CDX = X₂. Show that √X₁ + √X₂ ≤ √X, with equality iff AB is parallel to CD.
3. N is a positive integer > 2. Show that there are the same number of pairs of positive integers a < b ≤ N such that b/a > 2 and such that b/a < 2.
4. Show that the only solution to x + y² + z³ = 3, y + z² + x³ = 3, z + x² + y³ = 3 in positive reals is x = y = z = 1.
5. In an m x n array of reals the difference between the smallest and largest number in each row is at most d > 0. We now rearrange each column in decreasing order. Show that after the rearrangement the difference between the smallest and largest number in each row is still at most d.
6. A finite number of intervals cover [0, 1]. Show that one can find a subset of pairwise disjoint intervals with total length at least 1/2.

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