| 1. The polynomial a8x8 +a7x7 + ... + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0. |
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| 2. A unit square lies across two parallel lines a unit distance apart, so that two triangular areas of the square lie outside the lines. Show that the sum of the perimeters of these two triangles is independent of how the square is placed. |
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| 3. k > 14 is an integer and p is the largest prime smaller than k. k is chosen so that p ≥ 3k/4. Prove that 2p does not divide (2p - k)!, but that n does divide (n - k)! for composite n > 2p. |
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| 4. Show that (an + bn)1/n + (bn + cn)1/n + (cn + an)1/n < 1 + (21/n)/2, where n > 1 is an integer and a, b, c are the sides of a triangle with unit perimeter. |
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| 5. Find the smallest positive integer k such that among any k people, either there are 2m who can be divided into m pairs of people who know each other, or there are 2n who can be divided into n pairs of people who do not know each other. |
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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
28 April 2003
Last corrected/updated 28 Apr 03