| 1. Let an = 2n-1 for n > 0. Let bn = ∑r+s≤n aras. Find bn - bn-1, bn - 2bn-1 and bn. | |
| 2. Show that 1 - 1/k ≤ n(k1/n - 1) ≤ k - 1 for all positive integers n and positive reals k. |
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| 3. Let a1 = 1, a2 = 2a1, a3 = 3a2, a4 = 4a3, ... , a9 = 9a8. Find the last two digits of a9. |
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| 4. Find all polynomials p(x) such that p(x2) = p(x)2 for all x. Hence find all polynomials q(x) such that q(x2 - 2x) = q(x-2)2. | |
| 5. Find the smallest positive real t such that x1 + x3 = 2t x2, x2 + x4 = 2t x3, x3 + x5 = 2t x4 has a solution x1, x2, x3, x4, x5 in non-negative reals, not all zero. |
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| 6. For which n can we find positive integers a1, a2, ... , an such that a12 + a22 + ... + an2 is a square? |
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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
26 September 2003
Last corrected/updated 12 Feb 04