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1. Given two non-parallel lines e and f and a circle C which does not meet either line. Construct the line parallel to f such that the length of its segment inside C divided by the length of its segment from C to e (and outside C) is as large as possible.
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| 2. Let S(n) denote the sum of the decimal digits of the positive integer n. Find the set of all positive integers m such that s(km) = s(m) for k = 1, 2, ... , m. | |
| 3. Walking in the plane, we are allowed to move from (x, y) to one of the four points (x, y ± 2x), (x ± 2y, y). Prove that if we start at (1, √2), then we cannot return there after finitely many moves. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003