| 1. Find the side lengths of the triangle ABC with area S and ∠BAC = x such that the side BC is as short as possible. |
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| 2. Find all positive integers m, n such that n + (n+1) + (n+2) + ... + (n+m) = 1000. |
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| 3. Find a polynomial with integer coefficients which has √2 + √3 and √2 + 31/3 as roots. |
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| 4. Points H1, H2, ... , Hn are arranged in the plane so that each distance HiHj ≤ 1. The point P is chosen to minimise max(PHi). Find the largest possible value of max(PHi) for n = 3. Find the best upper bound you can for n = 4. | |
| 5. a1, a2, ... , an are constants such that f(x) = 1 + a1 cos x + a2 cos 2x + ... + an cos nx ≥ 0 for all x. We seek estimates of a1. If n = 2, find the smallest and largest possible values of a1. Find corresponding estimates for other values of n. |
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
23 September 2003
Last corrected/updated 9 Oct 03