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Not that I have seen, the theorems statement contains an undefined object...any 'proof' of it must involve some definitions so the meaningless collection of symbols e^{i*pi} is given some meaning. And taylor series are the standard way of defining what e^{something} is. (As well as what...

lols. Another topic that I got reminded of by a pm: Some people will have seen a 'proof' of Euler's formula before that claims to be within the mx2 syllabus. Here is why I think it is wrong. Originally Posted by bleakarcher Hey seanie, just a question. Would you consider this a satisfactory...

Not trying to defend incomplete reasoning here, but if we ignore the footnote then the solution now is pretty much enough. He proves both of the required inequalities and goes from there to saying that this implies k is the closest integer to sqrt(n). This is certainly not a fallacious argument...

haha its all good. by the way most of the things they ask you will have been asked before...if you ever get stuck just google the main parts of the problem and you will probably find something. you don't learn as much but its nice and fast.

Read this and adapt it http://mathforum.org/library/drmath/view/56653.html.

Assuming each box contains a token and each token is equally likely, the expected number of boxes needed is about 21.74. To do this, figure out the expected number of boxes needed to progress from owning k different tokens to owning (k+1) different tokens and add them up.

Indeed the footnote and the current solution are correct. My objection was the two lines quoted in my original post, P(k) is NOT necessarily maximised at the closest integer to root(n)+1/2, nor would this necessarily imply that P(k) is maximised at the integer closest to root(n). The key is...

He has fixed it now though.

Take n=5 then, same problem. My point was that the last step is not valid in general, and the square condition does not change this.

For that last part of the last question yeah, thats all I looked at. The integer closest to root(n) doesn't have to be the same as the integer closest to root(n)+1/2. Eg n=2.

Last two lines: P(k) is greatest when k is the integer closest to \sqrt{n}+1/2 so P(k) is greatest when k is the integer closest to \sqrt{n} Nice try terry.

Carrot is correct.

The reason I stress the cases in iv) is that it is very wrong to think that if alpha and beta are two distinct complex roots of a real quartic, then the roots are alpha, beta, conjugate of alpha, conjugate of beta. This is only necessarily the case if alpha and beta are NOT real. The reason why...

Well done on typing solutions Carrot :). I have just a couple of technical issues with the last two qn's that should not matter to most people: 15biii) How do we know that the real root must have multiplicity two? iv) Should probably consider the case where alpha is real, (although as we have...

Algebra question you may find interesting: Does there exist a unital ring R containing elements a,b such that ab is invertible but a and b are not?

Re: HSC 2012 Marathon :) Hint: polynomial degree. Gtg now but will look closer at your work later if you don't get it yet.

Yep, this is always the case. Luckily not everything in life is so "against the clock".

If P(k) < P(k-1) how can P(k) be the maximum of P? Sign issue. Also I am not sure how pedantic they are going to be about the 'closest integer' issue. For ciii) The thing you are given is equivalent to n > k^2 - k after squaring and simplifying. But if one integer is bigger than another, it...

I would think 0, but you will only know when BoS release their marking centre notes. These are the sort of pitfalls in Q15/16 I have been talking about, they are a lot sneakier than simply having a complicated integration by parts that you either get or you don't.

Ah right, yep. Good thing Carrotsticks asked it then!