5th British Mathematical Olympiad 1969 Problems

5th British Mathematical Olympiad 1969 Problems

1.  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.

2.  Find all real solutions to cos x + cos⁵x + cos 7x = 3.

3.  For which positive integers n can we find distinct integers a, b, c, d, a', b', c', d' greater than 1 such that n² - 1 = aa' + bb' + cc' + dd'? Give the solution for the smallest n.

4.  Find all integral solutions to a² - 3ab - a + b = 0.

5.  A long corridor has unit width and a right-angle 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?

6.  If a, b, c, d, e are positive integers, show that any divisor of both ae + b and ce + d also divides ad - bc.

7.  (1) f is a real-valued 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 real-valued function f on the reals, not identically zero, and differentiable at x = 0 which satisfies f(x) f(y) = f(x-y) for all x, y.

8.  A square side x has its vertices on the sides of a triangle with inradius r. Show that 2r > x > r√2.

9.  Let A_n be an n x n array of lattice points (n > 3). Is there a polygon with n² 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.

10.  Given a triangle, construct an equilateral triangle with the same area using ruler and compasses.