| 1. The circles OAB, OBC, OCA have equal radius r. Show that the circle ABC also has radius r. |
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| 2. Let x be a real number and put y = (x + 1)/2. Put an = 1 + x + x2 + ... + xn, and bn = 1 + y + y2 + ... + yn. Show that ∑0n am (n+1)C(m+1) = 2n bn, where aCb is the binomial coefficient a!/( b! (a-b)! ). |
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| 3. Show that an infinite arithmetic progression of unequal terms cannot consist entirely of primes. |
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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