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The Second British Mathematical Olympiad
1966年第二届英国奥林匹克数学竞赛
  1. Find the greatest and least values of f(x) = (x4 + x2 + 5)/(x2 + 1)2 for real x .
  2. For which distinct, real a, b, c are all the roots of ±√(x - a) ±√(x - b) ±√(x - c) = 0 real ?
  3. Sketch y2 = x2(x + 1)/(x - 1). Find all stationary values and describe the behaviour for large x .
  4. A1, A2, A3, A4 are consecutive vertices of a regular n-gon. 1/A1A2 = 1/A1A3 + 1/A1A4. What are the possible values of n ?
  5. A spanner has an enclosed hole which is a regular hexagon side 1. For what values of s can it turn a square nut side s ?
  6. Find the largest interval over which f(x) = √(x - 1) + √(x + 24 - 10√(x - 1) ) is real and constant .
  7. Prove that √2, √3 and √5 cannot be terms in an arithmetic progression .
  8. Given 6 different colours, how many ways can we colour a cube so that each face has a different colour? Show that given 8 different colours, we can colour a regular octahedron in 1680 ways so that each face has a different colour .
  9. The angles of a triangle are A, B, C. Find the smallest possible value of tan A/2 + tan B/2 + tan C/2 and the largest possible value of tan A/2 tan B/2 tan C/2 .
  10. One hundred people of different heights are arranged in a 10 x 10 array. X, the shortest of the 10 people who are the tallest in their row, is a different height from Y, the tallest of the 10 people who are the shortest in their column. Is X taller or shorter than Y ?
  11. (a) Show that given any 52 integers we can always find two whose sum or difference is a multiple of 100.
    (b) Show that given any set 100 integers, we can find a non-empty subset whose sum is a multiple of 100.
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