9th British Mathematical Olympiad 1973 Problems

9th British Mathematical Olympiad 1973 Problems

1.  A variable circle touches two fixed circles at P and Q. Show that the line PQ passes through one of two fixed points. State a generalisation to ellipses or conics.

2.  Given any nine points in the interior of a unit square, show that we can choose 3 which form a triangle of area at most 1/8.

3.  The curve C is the quarter circle x² + y² = r², x >= 0, y >= 0 and the line segment x = r, 0 >= y >= -h. C is rotated about the y-axis for form a surface of revolution which is a hemisphere capping a cylinder. An elastic string is stretched over the surface between (x, y, z) = (r sin θ, r cos θ, 0) and (-r, -h, 0). Show that if tan θ > r/h, then the string does not lie in the xy plane. You may assume spherical triangle formulae such as cos a = cos b cos c + sin b sin c cos A, or sin A cot B = sin c cot b - cos c cos A.

4.  n equilateral triangles side 1 can be fitted together to form a convex equiangular hexagon. The three smallest possible values of n are 6, 10 or 13. Find all possible n.

5.  Show that there is an infinite set of positive integers of the form 2ⁿ - 7 no two of which have a common factor.

6.  The probability that a teacher will answer a random question correctly is p. The probability that randomly chosen boy in the class will answer correctly is q and the probability that a randomly chosen girl in the class will answer correctly is r. The probability that a randomly chosen pupil's answer is the same as the teacher's answer is 1/2. Find the proportion of boys in the class.

7.  From each 10000 live births, tables show that y will still be alive x years later. y(60) = 4820 and y(80) = 3205, and for some A, B the curve Ax(100-x) + B/(x-40)² fits the data well for 60 <= x <= 100. Anyone still alive at 100 is killed. Find the life expectancy in years to the nearest 0.1 year of someone aged 70.

8.  T: z → (az + b)/(cz + d) is a map. M is the associated matrix

a  b

c  d

Show that if M is associated with T and M' with T' then the matrix MM' is associated with the map TT'. Find conditions on a, b, c, d for T⁴ to be the identity map, but T² not to be the identity map.

9.  Let L(θ) be the determinant:

x          y          1

a + c cos θ   b + c sin θ   1

l + n cos θ   m + n sin θ   1

Show that the lines are concurrent and find their point of intersection.

10.  Write a computer program to print out all positive integers up to 100 of the form a² - b² - c² where a, b, c are positive integers and a ≥ b + c.

11.  (1) A uniform rough cylinder with radius a, mass M, moment of inertia Ma²/2 about its axis, lies on a rough horizontal table. Another rough cylinder radius b, mass m, moment of inertia mb²/2 about its axis, rests on top of the first with its axis parallel. The cylinders start to roll. The plane containing the axes makes the angle θ with the vertical. Show the forces during the period when there is no slipping. Write down equations, which will give on elimination a differential equation for q, but you do not need to find the differential equation.

(2) Such a differential equation is θ₂(4 + 2 cos θ - 2 cos²θ + 9k/2) + θ₁² sin θ (2 cos θ - 1) = 3g(1 + k) (sin θ /(a + b), where k = M/m. Find θ₁ in terms of θ. Here θ₁ denotes dθ/dt and θ₂ denotes the second derivative.