The Fifth British Mathematical Olympiad
1969年第五届英国奥林匹克数学竞赛 
 Find the condition on the distinct real numbers a, b, c such that (x  a)(x  b)/(x  c) takes all real values. Sketch a graph where the condition is satisfied and another where it is not .
 Find all real solutions to cos x + cos^{5}x + cos 7x = 3 .
 For which positive integers n can we find distinct integers a, b, c, d, a', b', c', d' greater than 1 such that n^{2}  1 = aa' + bb' + cc' + dd'? Give the solution for the smallest n .
 Find all integral solutions to a^{2}  3ab  a + b = 0 .
 A long corridor has unit width and a rightangle corner. You wish to move a pipe along the corridor and round the corner. The pipe may have any shape, but every point must remain in contact with the floor. What is the longest possible distance between the two ends of the pipe ?
 If a, b, c, d, e are positive integers, show that any divisor of both ae + b and ce + d also divides ad  bc .
 (1) f is a realvalued function on the reals, not identically zero, and differentiable at x = 0. It satisfies f(x) f(y) = f(x+y) for all x, y. Show that f(x) is differentiable arbitrarily many times for all x and that if f(1) < 1, then f(0) + f(1) + f(2) + ... = 1/( 1  f(1) ) .
(2) Find the realvalued function f on the reals, not identically zero, and differentiable at x = 0 which satisfies f(x) f(y) = f(xy) for all x , y .
 A square side x has its vertices on the sides of a triangle with inradius r.
Show that 2r > x > r√2 .
 Let A_{n} be an n x n array of lattice points (n > 3). Is there a polygon with n^{2} sides whose vertices are the points of A_{n} such that no two sides intersect except adjacent sides at a vertex? You should prove the result for n = 4 and 5, but merely state why it is plausible for n > 5 .
 Given a triangle, construct an equilateral triangle with the same area using ruler and compasses .

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