| 1. Let p(x) = ax2 + bx + c be a quadratic with real coefficients. Show that we can find reals d, e, f so that p(x) = d/2 x(x - 1) + ex + f, and that p(n) is always integral for integral n iff d, e, f are integers. |
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| 2. P is a variable point outside the fixed sphere S with center O. Show that the surface area of the sphere center P radius PO which lies inside S is independent of P. |
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| 3. The triangle ABC has area k and angle A = θ, and is such that BC is as small as possible. Find AB and AC. |
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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 1, 1894-1905, 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