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The Fourteenth British Mathematical Olympiad
1978年第十四届英国奥林匹克数学竞赛
  1. Find the point inside a triangle which has the largest product of the distances to the three sides.
  2. Show that there is no rational number m/n with 0 < m < n < 101 whose decimal expansion has the consecutive digits 1, 6, 7 (in that order).
  3. Show that there is a unique sequence a1, a2, a3, ... such that a1 = 1, a2 > 1, an+1an-1 = an3 + 1, and all terms are integral.
  4. An altitude of a tetrahedron is a perpendicular from a vertex to the opposite face. Show that the four altitudes are concurrent iff each pair of opposite edges is perpendicular.
  5. There are 11000 points inside a cube side 15. Show that there is a sphere radius 1 which contains at least 6 of the points.
  6. Show that 2 cos nx is a polynomial of degree n in (2 cos x). Hence or otherwise show that if k is rational then cos kπ is 0, ±1/2, ±1 or irrational.
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