| 1. For what positive integers m, n can we find positive integers a, b, c such that a + mb = n and a + b = mc. Show that there is at most one such solution for each m, n. |
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| 2. Divide the unit square into 9 equal squares (forming a 3 x 3 array) and color the central square red. Now subdivide each of the 8 uncolored squares into 9 equal squares and color each central square red. Repeat n times, so that the side length of the smallest squares is 1/3n. How many squares are uncolored? What is the total red area as n → ∞? |
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| 3. ABC is a triangle and R any point on the segment AB. Let P be the intersection of the line BC and the line through A parallel to CR. Let Q be the intersection of the line AC and the line through B parallel to CR. Show that 1/AP + 1/BQ = 1/CR. |
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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