8th British Mathematical Olympiad 1972 Problems

8th British Mathematical Olympiad 1972 Problems

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 cosec θ.

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 a_n, by a₁ = 0, a₂ = 1, a₃= 2, a₄ = 3, and a_2n = a_2n-5 + 2ⁿ, a_2n+1 = a_2n + 2^n-1. Show that a_2n = [17/7 2^n-1] - 1, a_2n-1 = [12/7 2^n-1] - 1.

7.  Define sequences of integers by p₁ = 2, q₁ = 1, r₁ = 5, s₁= 3, p_n+1 = p_n² + 3 q_n², q_n+1 = 2 p_nq_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_nq_n²).

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.