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.

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