| 1. The roots of x3 - x - 1 = 0 are r, s, t. Find (1 + r)/(1 - r) + (1 + s)/(1 - s) + (1 + t)/(1 - t). |
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| 2. Find all real solutions to the equations x = 4z2/(1 + 4z2), y = 4x2/(1 + 4x2), z = 4y2/(1 + 4y2). |
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| 3. Let N be the number of permutations of 1, 2, 3, ... , 1996 in which 1 is fixed and each number differs from its neighbours by at most 2. Is N divisible by 3? |
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| 4. In the triangle ABC, AB = AC and the bisector of angle B meets AC at E. If BC = BE + EA find angle A. |
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| 5. Let x1, x2, ... , xm be positive rationals with sum 1. What is the maximum and minimum value of n - [n x1] - [n x2] - ... - [n xm] for positive integers n? |
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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
15 June 2002