The 24th British Mathematical Olympiad
1988年第24届英国奥林匹克数学竞赛 
 ABC is an equilateral triangle. S is the circle diameter AB. P is a point on AC such that the circle center P radius PC touches S at T. Show that AP/AC = 4/5. Find AT/AC .
 Show that the number of ways of dividing {1, 2, ... , 2n} into n sets of 2 elements is 1.3.5 ... (2n1). There are 5 married couples at a party. How many ways may the 10 people be divided into 5 pairs if no married couple may be paired together? For example, for 2 couples a, A, b, B the answer is 2: ab, AB; aB, bA .
 The real numbers a, b, c, x, y, z satisfy: x^{2}  y^{2}  z^{2} = 2ayz, x^{2} + y^{2}  z^{2} = 2bzx, x^{2}  y^{2} + z^{2} = 2cxy, and xyz ≠ 0. Show that x^{2}(1  b^{2}) = y^{2}(1  a^{2}) = xy(ab  c) and hence find a^{2} + b^{2} + c^{2}  2abc (independently of x, y, z) .
 Find all positive integer solutions to 1/a + 2/b  3/c = 1 .
 L and M are skew lines in space. A, B are points on L, M respectively such that AB is perpendicular to L and M. P, Q are variable points on L, M respectively such that PQ is of constant length. P does not coincide with A and Q does not coincide with B. Show that the center of the sphere through A, B, P, Q lies on a fixed circle whose center is the midpoint of AB .
 Show that if there are triangles with sides a, b, c, and A, B, C, then there is also a triangle with sides √(a^{2} + A^{2}), √(b^{2} + B^{2}), √(c^{2} + C^{2}) .

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