4th British Mathematical Olympiad 1968 Problems

4th British Mathematical Olympiad 1968 Problems 

1.  C is the circle center the origin and radius 2. Another circle radius 1 touches C at (2, 0) and then rolls around C. Find equations for the locus of the point P of the second circle which is initially at (2, 0) and sketch the locus.

2.  Cows are put in a field when the grass has reached a fixed height, any cow eats the same amount of grass a day. The grass continues to grow as the cows eat it. If 15 cows clear 3 acres in 4 days and 32 cows clear 4 acres in 2 days, how many cows are needed to clear 6 acres in 3 days? 

3.  The distance between two points (x, y) and (x', y') is defined as |x - x'| + |y - y'|. Find the locus of all points with non-negative x and y which are equidistant from the origin and the point (a, b) where a > b.

4.  Two balls radius a and b rest on a table touching each other. What is the radius of the largest sphere which can pass between them?

5.  If reals x, y, z satisfy sin x + sin y + sin z = cos x + cos y + cos z = 0. Show that they also satisfy sin 2x + sin 2y + sin 2z = cos 2x + cos 2y + cos 2z = 0.

6.  Given integers a₁, a₂, ... , a₇ and a permutation of them a_f(1), a_f(2), ... , a_f(7), show that the product (a₁ - a_f(1))(a₂ - a_f(2)) ... (a₇ - a_f(7)) is always even.

7.  How many games are there in a knock-out tournament amongst n people?

8.  C is a fixed circle of radius r. L is a variable chord. D is one of the two areas bounded by C and L. A circle C' of maximal radius is inscribed in D. A is the area of D outside C'. Show that A is greatest when D is the larger of the two areas and the length of L is 16πr/(16 + π²).

9.  The altitudes of a triangle are 3, 4, 6. What are its sides?

10.  The faces of the tetrahedron ABCD are all congruent. The angle between the edges AB and CD is x. Show that cos x = sin(∠ABC - ∠BAC)/sin(∠ABC + ∠BAC).

11.  The sum of the reciprocals of n distinct positive integers is 1. Show that there is a unique set of such integers for n = 3. Given an example of such a set for every n > 3.

12.  What is the largest number of points that can be placed on a spherical shell of radius 1 such that the distance between any two points is at least √2? What is the largest number such that the distance is > √2?