| 1. Show that the quadratic x2 + 2mx + 2n has no rational roots for odd integers m, n. |
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| 2. Let r be the radius of a circle through three points of a parallelogram. Show that any point inside the parallelogram is a distance ≤ r from at least one of its vertices. |
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| 3. Show that the decimal expansion of a rational number must repeat from some point on. [In other words, if the fractional part of the number is 0.a1a2a3 ... , then an+k = an for some k > 0 and all n > some n0.] |
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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