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 q are coprime.

A2.  P is a point inside the triangle ABC. Angle PAC = angle PBC. M is the midpoint of AB and L and N are the feet of the perpendiculars from P to BC and CA respectively. Show that ML = MN.

A3.  A box contains w white and b black balls. Two balls taken at random are removed. If they are the same color, then a black ball is put into the box. If they are the opposite color, then a white ball is put into the box. This is repeated until the box contains only one ball. What is the probability that it is white?

B1.  Find all positive integers m, n such that (n+1)^m = n! + 1.

B2.  Find the permutations a₁, a₂, ... , a_n of 1, 2, ... , n which maximise and minimise a₁a₂ + a₂a₃ + ... + a_n-1a_n + a_na₁.

B3.  ABC is right-angled and similar to AB'C'. But ABC and AB'C' have opposite orientation. The right-angles are at B and B'. BC' and B'C meet at X. Show that AX is perpendicular to BB'.