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The Seventeenth British Mathematical Olympiad
1981年第十七届英国奥林匹克数学竞赛
  1. ABC is a triangle. Three lines divide the triangle into four triangles and three pentagons. One of the triangle has its three sides along the new lines, the others each have just two sides along the new lines. If all four triangles are congruent, find the area of each in terms of the area of ABC .
  2. An axis of a solid is a straight line joining two points on its boundary such that a rotation about the line through an angle greater than 0 deg and less than 360 deg brings the solid into coincidence with itself. How many such axes does a cube have? For each axis indicate the minimum angle of rotation and how the vertices are permuted .
  3. Find all real solutions to x2y2 + x2z2 = axyz, y2z2 + y2x2 = bxyz, z2x2 + z2y2 = cxyz, where a, b, c are fixed reals .
  4. Find the remainder on dividing x81 + x49 + x25 + x9 + x by x3 - x.
  5. The sequence u0, u1, u2, ... is defined by u0 = 2, u1 = 5, un+1un-1 - un2 = 6n-1. Show that all terms of the sequence are integral .
  6. Show that for rational c, the equation x3 - 3cx2 - 3x + c = 0 has at most one rational root.
  7. If x and y are non-negative integers, show that there are non-negative integers a, b, c, d such that x = a + 2b + 3c + 7d, y = b + 2c + 5d iff 5x ≥ 7y .
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