The 22nd British Mathematical Olympiad
1986年第22届英国奥林匹克数学竞赛 
 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 .
 Find the maximum value of x^{2}y + y^{2}z + z^{2}x for reals x, y, z with sum zero and sum of squares 6 .
 P_{1}, P_{2}, ... , P_{n} are distinct subsets of {1, 2, ... , n} with two elements. Distinct subsets P_{i} and P_{j} have an element in common iff {i, j} is one of the P_{k}. Show that each member of {1, 2, ... , n} belongs to just two of the subsets .
 m ≤ n are positive integers. nCm denotes the binomial coefficient n!/(m! (nm)! ). Show that nCm nC(m1) is divisible by n. Find the smallest positive integer k such that k nCm nC(m1) nC(m2) is divisible by n^{2} 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(m1) nC(m2) )/n^{2} where 1 < m ≤ n .
 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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