| 1. Circles C and C' meet at A and B. The arc AB of C' divides the area inside C into two equal parts. Show that its length is greater than the diameter of C. |
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| 2. a, b, c are reals such that |ax2 + bx + c| ≤ 1 for all x ≤ |1|. Show that |2ax + b| ≤ 4 for all |x| ≤ 1. |
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| 3. A circle meets the side BC of the triangle ABC at A1, A2. Similarly, it meets CA at B1, B2, and it meets AB at C1, C2. The perpendicular to BC at A1, the perpendicular to CA at B1 and the perpendicular to AB at C1 meet at a point. Show that the perpendiculars at A2, B2, C2 also meet at a point. |
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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 31 Oct 03