8th Australian Mathematical Olympiad Problems 1987

A1.  ABC is an isosceles triangle with AB = AC. M is the midpoint of AC. D is a point on the arc BC of the circumcircle of BMC not containing M, and the ray BD meets the ray AC at E so that DE = MC. Show that MD2 = AC·CE and CE2 = BC·MD/2. A2.  Show that (2p)!/(p! p!) - 2 is a multiple of p if p is prime. A3.  A graph has 20 points and there is an edge between

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7th Australian Mathematical Olympiad Problems 1986

A1.  Given a positive integer n and real k > 0, what is the largest possible value for (x1x2 + x2x3 + x3x4 + ... + xn-1xn), where xi are non-negative real numbers with sum k? A2.  What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a1 = 3, a2 = 33, and an+1 = 3an. Similarly, b1 = 100, b2 = 100100 etc. What is the smallest n for which bn >

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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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4th Australian Mathematical Olympiad Problems 1983

A1.  Consider the following sequence: 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, ... , where we list all m/n with m+n = k in order of decreasing m, and then all m/n with m+n = k+1 etc. Each rational appears many times. Find the first five positions of 1/2. What is the position for the nth occurrence of 1/2? Find an expression for the first occurrence of p/q where p < q and p and

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3rd Australian Mathematical Olympiad Problems 1982

A1.  If you toss a fair coin n+1 times and I toss it n times, what is the probability that you get more heads? A2.  Show that the fractional part of (2 + √3)n tends to 1. A3.  In the triangle ABC, let the angle bisectors of A, B, C meet the circumcircle again at X, Y, Z. Show that AX + BY + CZ is greater than the perimeter of ABC. B1.  For what d

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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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1st Australian Mathematical Olympiad Problems 1979

1.  A graph with 10 points and 35 edges is constructed as follows. Every vertex of one pentagon is joined to every edge of another pentagon. Each edge is colored black or white, so that there are no monochrome triangles. Show that all 10 edges of the two pentagons have the same color. 2.  Two circles (not necessarily equal) intersect at A and B. A point P travels clockwise

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