My current take on the question would be to prove it through induction, but it might not yield any result.
It would be best to train yourself for the AMO not by doing past questions straight away if you are not familiar with it. AMT does offer several books on how to tackle olympiad style questions, whether they are effective I am not sure.
Have you done the AMOC Correspondence Program 4 last year? It is used to give people some practice at solving olympiad style questions, by providing hints to guide you. Certainly easier than the AMO, but it is something to start with.
As for your PM, I'll answer it here just in case there is anyone else that is after the same information.
The 2006 AMO will be held on the 7th and 8th Feb. Each day consists of 4 questions, with 4 hours to do them. Like standard international olympiads each question is marked out of 7, and the marks are gained exponentially (ie relatively easy to get 1 mark, harder to get the next mark etc).
There isn't much specific advice I can give you beyond what AMT recommends, since the questions can cover quite a wide range of topics. The most important thing is to realise that you do have all those time, and it's not uncommon to be stuck on a question and not realise any possible ways to solve it until later. Keep on thinking about them.
A useful tip is to draw large sized, proper scaled diagram for geometric questions. It might not give you the answer (or the answer but no working), but can provide you insights into how to solve it. Or in the case of giving you the answer, you can work from both ends to meet in the middle, or know what you're looking out for when writing out your working.
Write every idea you have on the question down and try to justify everything. You never know what you might be given a mark for.
If this question seems very different, don't stress out, because it is certainly one of the harder ones. However, even if you do find the questions in the AMO intimidating, don't worry either, you're already within an elite group of people in Australia to participate in the AMO (roughly 100 or so a year).
That's pretty much all I can think of at the moment. SeDaTeD did the AMO in 2003 and got a bronze, so he might have some inputs of his own.