2nd Australian Mathematical Olympiad Problems 1981

A1.  Show that in any set of 27 distinct odd positive numbers less than 100 we can always find two with sum 102. How many sets of 26 odd positive numbers less 100 than can we find with no two having sum 102? A2.  Given a real number 0 < k < 1, define p(x) = (x - k)(x - k2) ... (x - kn)/( (x + k)(x + k2) ... (x + kn) ). Show that if kn+1 ≤ x < 1, then p(x) < p(1).

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