The Eighth British Mathematical Olympiad
1972年第八届英国奥林匹克数学竞赛 
 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 ?
 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 .
 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 .
 P is a point on a curve through A and B such that PA = a, PB = b, AB = c, and ∠APB = θ. As usual, c^{2} = a^{2} + b^{2}  2ab cos θ. Show that sin^{2}θ ds^{2} = da^{2} + db^{2}  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.
 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 semivertical angle x. Show that if x is varied the maximum value of vol C/vol R is at sin x = 1/3.
 Define the sequence a_{n}, by a_{1} = 0, a_{2} = 1, a_{3}= 2, a_{4} = 3, and a_{2n} = a_{2n5} + 2^{n}, a_{2n+1} = a_{2n} + 2^{n1}. Show that a_{2n} = [17/7 2^{n1}]  1, a_{2n1} = [12/7 2^{n1}]  1.
 Define sequences of integers by p_{1} = 2, q_{1} = 1, r_{1} = 5, s_{1}= 3, p_{n+1} = p_{n}^{2} + 3 q_{n}^{2}, q_{n+1} = 2 p_{n}q_{n}, r_{n} = p_{n} + 3 q_{n}, s_{n} = p_{n} + q_{n}. Show that p_{n}/q_{n} > √3 > r_{n}/s_{n} and that p_{n}/q_{n} differs from √3 by less than s_{n}/(2 r_{n}q_{n}^{2}) .
 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.
 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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