| 1. For which positive integers n does 3 divide 2n + 1? |
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| 2. Triangles ABC, PQR satisfy (1) ∠A = ∠P, (2) |∠B - ∠C| < |∠Q - ∠R|. Show that sin A + sin B + sin C > sin P + sin Q + sin R. What angles A, B, C maximise sin A + sin B + sin C? |
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| 3. The line L contains the distinct points A, B, C, D in that order. Construct a square such that two opposite sides (or their extensions) intersect L at A, B, and the other two sides (or their extensions) intersect L at C, D. |
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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
Last corrected/updated 30 Oct 03