| 1. n and 3n have the same digit sum. Show that n is a multiple of 9. |
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| 2. A railway line passes through (in order) the 11 stations A, B, ... , K. The distance from A to K is 56. The distances AC, BD, CE, ... , IK are each ≤ 12. The distances AD, BE, CF, ... , HK are each ≥ 17. Find the distance between B and G. |
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| 3. Show that if ab are integers with ab even, then a2 + b2 + x2 = y2 has an integral solution x, y. |
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| 4. * is a real-valued operation on the non-zero reals which satisfies (1) a * a = 1, (2) a * (b * c) = (a * b) c (in other words, the ordinary product of (a * b) and c), for all a, b, c. Solve x * 36 = 216. | |
| 5. Given a triangle with sides a, b, c, we form, if we can, a triangle with sides s-a, s-b, s-c, where s = (a+b+c)/2. For which triangles can this be repeated indefinitely? |
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| 6. For reals a, b define the function f(x) = 1/(ax+b). For which a, b are there distinct reals x1, x2, x3 such that f(x1) = x2, f(x2) = x3, f(x3) = x1. |
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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
9 October 2003
Last corrected/updated 17 Jan 04