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The Eighth British Mathematical Olympiad
1972年第八届英国奥林匹克数学竞赛
  1. The relation R is defined on the set X. It has the following two properties: if aRb and bRc then cRa for distinct elements a, b, c; for distinct elements a, b either aRb or bRa but not both. What is the largest possible number of elements in X ?
  2. Show that there can be at most four lattice points on the hyperbola (x + ay + c)(x + by + d) = 2, where a, b, c, d are integers. Find necessary and sufficient conditions for there to be four lattice points .
  3. C and C' are two unequal circles which intersect at A and B. P is an arbitrary point in the plane. What region must P lie in for there to exist a line L through P which contains chords of C and C' of equal length. Show how to construct such a line if it exists by considering distances from its point of intersection with AB or otherwise .
  4. P is a point on a curve through A and B such that PA = a, PB = b, AB = c, and ∠APB = θ. As usual, c2 = a2 + b2 - 2ab cos θ. Show that sin2θ ds2 = da2 + db2 - 2 da db cos θ, where s is distance along the curve. P moves so that for time t in the interval T/2 < t < T, PA = h cos(t/T), PB = k sin(t/T). Show that the speed of P varies as cosecθ.
  5. A cube C has four of its vertices on the base and four of its vertices on the curved surface of a right circular cone R with semi-vertical angle x. Show that if x is varied the maximum value of vol C/vol R is at sin x = 1/3.
  6. Define the sequence an, by a1 = 0, a2 = 1, a3= 2, a4 = 3, and a2n = a2n-5 + 2n, a2n+1 = a2n + 2n-1. Show that a2n = [17/7 2n-1] - 1, a2n-1 = [12/7 2n-1] - 1.
  7. Define sequences of integers by p1 = 2, q1 = 1, r1 = 5, s1= 3, pn+1 = pn2 + 3 qn2, qn+1 = 2 pnqn, rn = pn + 3 qn, sn = pn + qn. Show that pn/qn > √3 > rn/sn and that pn/qn differs from √3 by less than sn/(2 rnqn2) .
  8. Three children throw stones at each other every minute. A child who is hit is out of the game. The surviving player wins. At each throw each child chooses at random which of his two opponents to aim at. A has probability 3/4 of hitting the child he aims at, B has probability 2/3 and C has probability 1/2. No one ever hits a child he is not aiming at. What is the probability that A is eliminated in the first round and C wins.
  9. A rocket, free of external forces, accelerates in a straight line. Its mass is M, the mass of its fuel is m exp(-kt) and its fuel is expelled at velociy v exp(-kt). If m is small compared to M, show that its terminal velocity is mv/(2M) times its initial velocity .
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