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The 24th British Mathematical Olympiad
1988年第24届英国奥林匹克数学竞赛
  1. 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 .
  2. Show that the number of ways of dividing {1, 2, ... , 2n} into n sets of 2 elements is 1.3.5 ... (2n-1). 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 .
  3. The real numbers a, b, c, x, y, z satisfy: x2 - y2 - z2 = 2ayz, -x2 + y2 - z2 = 2bzx, -x2 - y2 + z2 = 2cxy, and xyz ≠ 0. Show that x2(1 - b2) = y2(1 - a2) = xy(ab - c) and hence find a2 + b2 + c2 - 2abc (independently of x, y, z) .
  4. Find all positive integer solutions to 1/a + 2/b - 3/c = 1 .
  5. 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 .
  6. Show that if there are triangles with sides a, b, c, and A, B, C, then there is also a triangle with sides √(a2 + A2), √(b2 + B2), √(c2 + C2) .
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