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The Nineteenth British Mathematical Olympiad
1983年第十九届英国奥林匹克数学竞赛
  1. Given points A and B and a line l, find the point P which minimises PA2 + PB2 + PN2, where N is the foot of the perpendicular from P to l. State without proof a generalisation to three points .
  2. Each pair of excircles of the triangle ABC has a common tangent which does not contain a side of the triangle. Show that one such tangent is perpendicular to OA, where O is the circumcenter of ABC .
  3. l, m, and n are three lines in space such that neither l nor m is perpendicular to n. Variable points P on l and Q on m are such that PQ is perpendicular to n, The plane through P perpendicular to m meets n at R, and the plane through Q perpendicular to l meets n at S. Show that RS has constant length .
  4. Show that for any positive reals a, b, c, d, e, f we have ab/(a + b) + cd/(c + d) + ef/(e + f) ≤ (a + c + e)(b + d + f)/(a + b + c + d + e + f) .
  5. How many permutations a, b, c, d, e, f, g, h of 1, 2, 3, 4, 5, 6, 7, 8 satisfy a < b, b > c, c < d, d> e, e < f, f > g, g < h ?
  6. Find all positive integer solutions to (n + 1)m = n! + 1 .
  7. Show that in a colony of mn + 1 mice, either there is a set of m + 1 mice, none of which is a parent of another, or there is an ordered set of n + 1 mice (M0, M1, M2, ... , Mn) such that Mi is the parent of Mi-1 for i = 1, 2, ... , n .
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