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The First British Mathematical Olympiad
1965年第一届英国奥林匹克数学竞赛
  1. Sketch f(x) = (x2 + 1)/(x + 1). Find all points where f '(x) = 0 and describe the behaviour when x or f(x) is large.
  2. X, at the centre a circular pond. Y, at the edge, cannot swim, but can run at speed 4v. X can run faster than 4v and can swim at speed v. Can X escape ?
  3. Show that np - n is divisible by p for p = 3, 7, 13 and any integer n.
  4. What is the largest power of 10 dividing 100 x 99 x 98 x ... x 1 ?
  5. Show that n(n + 1)(n + 2)(n + 3) + 1 is a square for n = 1 , 2 , 3 , ... .
  6. The fractional part of a real is the real less the largest integer not exceeding it. Show that we can find n such that the fractional part of (2 + √2)n > 0.999 .
  7. What is the remainder on dividing x + x3 + x9 + x27 + x81 + x243 by x - 1? By x2 - 1 ?
  8. For what real b can we find x satisfying: x2 + bx + 1 = x2 + x + b = 0 ?
  9. Show that for any real, positive x, y, z, not all equal, we have: (x + y)(y + z)(z + x) > 8 xyz.
  10. A chord length √3 divides a circle C into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of rectangle than can be drawn in R ?
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