26th Swedish 1986

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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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© John Scholes
jscholes@kalva.demon.co.uk
8 October 2003
Last corrected/updated 27 Feb 04