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The 22nd British Mathematical Olympiad
1986年第22届英国奥林匹克数学竞赛
  1. A rational point is a point both of whose coordinates are rationals. Let A, B, C, D be rational points such that AB and CD are not both equal and parallel. Show that there is just one point P such that the triangle PCD can be obtained from the triangle PAB by enlargement and rotation about P. Show also that P is rational .
  2. Find the maximum value of x2y + y2z + z2x for reals x, y, z with sum zero and sum of squares 6 .
  3. P1, P2, ... , Pn are distinct subsets of {1, 2, ... , n} with two elements. Distinct subsets Pi and Pj have an element in common iff {i, j} is one of the Pk. Show that each member of {1, 2, ... , n} belongs to just two of the subsets .
  4. m ≤ n are positive integers. nCm denotes the binomial coefficient n!/(m! (n-m)! ). Show that nCm nC(m-1) is divisible by n. Find the smallest positive integer k such that k nCm nC(m-1) nC(m-2) is divisible by n2 for all m, n such that 1 < m ≤ n. For this value of k and fixed n, find the greatest common divisor of the n - 1 integers ( k nCm nC(m-1) nC(m-2) )/n2 where 1 < m ≤ n .
  5. C and C' are fixed circles. A is a fixed point on C, and A' is a fixed point on C'. B is a variable point on C. B' is the point on C' such that A'B' is parallel to AB. Find the locus of the midpoint of BB' .
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