6th Australian Mathematical Olympiad Problems 1985

A1.  Find the sum of the first n terms of 0, 1, 1, 2, 2, 3, 3, 4, 4, ... (each positive integer occurs twice). Show that the sum of the first m + n terms is mn larger than the sum of the first m - n terms. A2.  Show that any real values satisfying x + y + z = 5, xy + yz + zx = 3 lie between -1 and 13/3. A3.  A graph has 9 points and 36 edges. Each edge is

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