The Fourth British Mathematical Olympiad
1968年第四届英国奥林匹克数学竞赛 
 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.
 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 ?
 The distance between two points (x, y) and (x', y') is defined as x  x' + y  y'. Find the locus of all points with nonnegative x and y which are equidistant from the origin and the point (a, b) where a > b.
 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 ?
 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.
 Given integers a_{1}, a_{2}, ... , a_{7} and a permutation of them a_{f(1)} , a_{f(2)} , ... , a_{f(7) }, show that the product (a_{1}  a_{f(1)})(a_{2}  a_{f(2)}) ... (a_{7}  a_{f(7)}) is always even.
 How many games are there in a knockout tournament amongst n people ?
 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 + π^{2}).
 The altitudes of a triangle are 3, 4, 6. What are its sides ?
 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).
 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.
 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 ?

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