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The 25th British Mathematical Olympiad
1989年第25届英国奥林匹克数学竞赛
  1. Find the smallest positive integer a such that ax2 - bx + c = 0 has two distinct roots in the interval 0 < x < 1 for some integers b , c .
  2. Find the number of different ways of arranging five As, five Bs and five Cs in a row so that each letter is adjacent to an identical letter. Generalise to n letters each appearing five times .
  3. f(x) is a polynomial of degree n such that f(0) = 0, f(1) = 1/2, f(2) = 2/3, f(3) = 3/4, ... , f(n) = n/(n+1). Find f(n+1) .
  4. D is a point on the side AC of the triangle ABC such that the incircles of BAD and BCD have equal radii. Express | BD | in terms of the lengths a = | BC |, b = | CA |, c = | AB | .
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