| 1. α, β, γ are real and satisfy α2 + β2 + γ2 = 1. Show that -1/2 ≤ αβ + βγ + γα ≤ 1. |
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| 2. If ac, bc + ad, bd = 0 (mod n) show that bc, ad = 0 (mod n). |
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| 3. ABC is a triangle with angle C = 120o. Find the length of the angle bisector of angle C in terms of BC and CA. |
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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