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The Sixteenth British Mathematical Olympiad
1980年第十六届英国奥林匹克数学竞赛
  1. Show that there are no solutions to an + bn = cn, with n > 1 is an integer, and a, b, c are positive integers with a and b not exceeding n .
  2. Find a set of seven consecutive positive integers and a polynomial p(x) of degree 5 with integer coefficients such that p(n) = n for five numbers n in the set including the smallest and largest, and p(n) = 0 for another number in the set .
  3. AB is a diameter of a circle. P, Q are points on the diameter and R, S are points on the same arc AB such that PQRS is a square. C is a point on the same arc such that the triangle ABC has the same area as the square. Show that the incenter I of the triangle ABC lies on one of the sides of the square and on the line joining A or B to R or S .
  4. Find all real a0 such that the sequence a0, a1, a2, ... defined by an+1 = 2n - 3an has an+1 > an for all n ≥ 0 .
  5. A graph has 10 points and no triangles. Show that there are 4 points with no edges between them .
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