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The Fourth British Mathematical Olympiad
1968年第四届英国奥林匹克数学竞赛
  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 af(1) , af(2) , ... , af(7) , show that the product (a1 - af(1))(a2 - af(2)) ... (a7 - af(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 ?
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