| 1. d is not divisible by 5. For some integer n, a n3 + b n2 + c n + d is divisible by 5. Show that for some integer m, a + b m + c m2 + d m3 is divisible by 5. |
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| 2. Construct the triangle ABC given c, r and r', where c = |AB|, r is the radius of the inscribed circle, and r' is the radius of the other circle tangent to the segment AB and the lines BC and CA. |
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| 3. Two particles fall from rest 300 m under the influence of gravity alone. One particle leaves when the other has already fallen 1 μm. How far apart are they when the first particle reaches the end point (to the nearest 100 μm)? |
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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