| 1. For a positive integer n, let p(n) be the number of prime factors of n. Show that ln n ≥ p(n) ln 2. |
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| 2. Show that if (a, b) satisfies a2 - 3ab + 2b2 + a - b = a2 - 2ab + b2 - 5a + 7b = 0, then it also satisfies ab - 12a + 15b = 0. |
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| 3. Given three points P, Q, R in the plane, find points A, B, C such that P is the foot of the perpendicular from A to BC, Q is the foot of the perpendicular from B to CA, and R is the foot of the perpendicular from C to AB. Find the lengths AB, BC, CA in terms of PQ, QR and RP. |
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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
Last corrected/updated 6 Jan 03