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Yeah these are the ones that I wrote. (Carrot wrote the other half.) I have not made any changes.

Some are tricky, but I took care to break them down into doable steps. I scrapped many harder questions :). No worries.

Attached is a handful of somewhat difficult MX2 level questions written by me for a BoS meat a while back. Some of you will have seen them before, but I never wrote up solutions here (and don't really have the time to do so now). If you think you have solved any of them feel free to post here...

By convention the product of an empty collection of numbers is 1 and the sum of an empty collection of numbers is 0. I stress that this is just convention, it is convenient for this to be the definition.

*non-positive integers*

It is an easy recurrence to solve... A_{2k}=(-1)^k\frac{1}{2^kk!},\quad A_{2k+1}=\frac{(-1)^k}{1\cdot3\cdot\ldots\cdot(2k+1)}=\frac{(-2)^kk!}{(2k+1)!}. And I would recommend doing this (solving the recurrence) whenever possible in such questions.

Glad you like it :). Of course, its sexiness is slighly detracted from by the fact that it is very easy to EXPLICITLY construct a pair: \sqrt{3}^{2\log_3{2}}=2 (Some mathematicians object to nonconstructive proofs.) But I still think its a pretty striking argument.

Carrot, have you ever seen the really sexy nonconstructive proof that you can find irrationals a,b such that a^b is rational?

Yeah that guy is very clever.

Aim for it, sure. Though they tend to be quite anal in the marking, the last perfect score I know of was by Anthony Henderson in 1993.

Interesting question with a pretty (I think) elegant answer: Do there exist a pair of irrational numbers a and b such that a^b is rational? Hint: Consider \sqrt{2}^\sqrt{2}.

Re: Math help (Because Carrot made me do this) To establish the validity of these "modern techniques" one has to write considerably more than 5 lines. There are proofs that do not require so much machinery. Eg Gauss' doctoral proof of FTA wasn't completely justified, he appealed a bit to...

Re: Math help (Because Carrot made me do this) I honestly would be happy to in topics other than graphing, but graphing is so hard to convey when not in person, it would be easier to get someone like Carrot or actually SD to help you out.

Re: Math help (Because Carrot made me do this) Yep. The graph I posted isn't too hard, but my point is anything that oscillates a lot makes plotting a few points next to useless.

Memorising things like that isn't necessarily distinct from studying 'smart'. In this case, it is a timesaving measure. But it is useful have a rough idea of how we could prove it.

This. Study smart not hard.

Re: Math help (Because Carrot made me do this) Kind of, but we would technically need to prove that no pair of these equations are redundant or contradictory, which isn't obvious. A more transparent way of proving this is by trying to construct the polynomial explicitly. EDIT. I strongly...

Re: Math help (Because Carrot made me do this) Provided that no pair of these points have the same x coordinate, but different y coordinates of course.

Still though, doubling the secondbest mark is making a pretty strong statement...I never managed that. The best I ever did vs cohort in prelim 3u was 100 in the yearlies, second was ~80, 3rd was ~70 and the rest failed, with average <40.

Epic man, congrats!