| 1. a, b, c, d are each relatively prime to n = ad - bc, and r and s are integers. Show ar + bs is a multiple of n iff cr + ds is a multiple of n. |
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| 2. Find the sum of all four digit numbers (written in base 10) which contain only the digits 1 - 5 and contain no digit twice. |
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| 3. r is the inradius of the triangle ABC and r' is the exradius for the circle touching AB. Show that 4r r' ≤ c2, where c is the length of the side AB. |
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The original problems are in Hungarian. They are available on the KöMaL archive on the web. They are also available in English (with solutions in): (Translated by Elvira Rapaport) József Kürschák, G Hajós, G Neukomm & J Surányi, Hungarian Problem Book 2, 1906-1928, MAA 1963. Out of print, but available in some university libraries.
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John Scholes
jscholes@kalva.demon.co.uk
20 Oct 1999
Last corrected/updated 1 Nov 03