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The Eighteenth British Mathematical Olympiad
1982年第十八届英国奥林匹克数学竞赛
  1. ABC is a triangle. The angle bisectors at A, B, C meet the circumcircle again at P, Q , R respectively. Show that AP + BQ + CR > AB + BC + CA .
  2. The sequence p1, p2, p3, ... is defined as follows. p1 = 2. pn+1 is the largest prime divisor of p1p2 ... pn + 1. Show that 5 does not occur in the sequence .
  3. a is a fixed odd positive integer. Find the largest positive integer n for which there are no positive integers x, y, z such that ax + (a + 1)y + (a + 2)z = n .
  4. a and b are positive reals and n > 1 is an integer. P1 (x1, y1) and P2 (x2, y2) are two points on the curve xn - ayn = b with positive real coordinates. If y1 < y2 and A is the area of the triangle OP1P2, show that  by2 > 2ny1n-1a1-1/nA .
  5. p(x) is a real polynomial such that p(2x) = 2k-1(p(x) + p(x + 1/2) ), where k is a non-negative integer. Show that p(3x) = 3k-1(p(x) + p(x + 1/3) + p(x + 2/3) ) .
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