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The Twelfth British Mathematical Olympiad
1976年第十二届英国奥林匹克数学竞赛
  1. ABC is a triangle area k. Let d be the length of the shortest line segment which bisects the area of the triangle. Find d. Give an example of a curve which bisects the area and has length < d.
  2. Prove that x/(y + z) + y/(z + x) + z/(x + y) ≥ 3/2 for any positive reals x, y, z.
  3. Given 50 distinct subsets of a finite set X, each containing more than | X |/2 elements, show that there is a subset of X with 5 elements which has at least one element in common with each of the 50 subsets.
  4. Show that 8n19 + 17 is not prime for any non-negative integer n.
  5. aCb represents the binomial coefficient a!/( (a - b)! b! ). Show that for n a positive integer, r ≤ n and odd, r' = (r - 1)/2 and x, y reals we have: ∑0r' nC(r-i) nCi (xr-iyi + xiyr-i) = ∑0r' nC(r-i) (r-i)Ci xiyi(x + y)r-2i.
  6. A sphere has center O and radius r. A plane p, a distance r/2 from O, intersects the sphere in a circle C center O'. The part of the sphere on the opposite side of p to O is removed. V lies on the ray OO' a distance 2r from O'. A cone has vertex V and base C, so with the remaining part of the sphere it forms a surface S. XY is a diameter of C. Q is a point of the sphere in the plane through V, X and Y and in the plane through O parallel to p. P is a point on VY such that the shortest path from P to Q along the surface S cuts C at 45 deg. Show that VP = r√3 / √(1 + 1/√5).
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