| 1. ABC is a right-angled triangle. Show that sin A sin B sin(A - B) + sin B sin C sin(B - C) + sin C sin A sin(C - A) + sin(A - B) sin(B - C) sin(C - A) = 0. |
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| 2. ABC is an arbitrary triangle. Show that sin(A/2) sin(B/2) sin(C/2) < 1/4. |
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| 3. The line L contains the distinct points A, B, C, D in that order. Construct a rectangle whose sides (or their extensions) intersect L at A, B, C, D and such that the side which intersects L at C has length k. How many such rectangles are there? |
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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