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Unless we write something silly, like: \frac{a}{b-c}=\frac{6a}{\pi^2(b-c)}\prod_{p\textrm{ prime}}(1-p^{-2})^{-1}. Where the infinite product is just a constant.

haha. its actually probably the best answer to give. Most other expressions in terms of a,b,c could probably be made to diverge for suitable choices of a.b,c.

My handwriting is fast but atrocious unless I am writing something for someone else / writing up a completed proof. Then its slow and legible. My own handwritten notes, (especially mathematical!) jump all over the place wherever I can find space on the page haha. Thankfully I can latex...

Never been tutored, I prefer learning things my own way rather than someone elses.

Re: 2012 HSC MX2 Marathon Don't let the HSC course make induction seem like a triviality, it is often a powerful way to prove results concisely. And indeed most non-obvious results are "guessed" well before they are proven. By the way, I don't entirely follow what you are saying in the early...

Re: 2012 HSC MX2 Marathon The induction is straightforward. $For $n=2$, $LHS-RHS=a_1a_2>0$. If we suppose that:\\ $\prod_{k=1}^{n-1}(1-a_k)>1-\sum_{k=1}^{n-1}a_k$ for all sequences $(a_k)$ with $0<a_j<1$ for all $j$, then:\\$\prod_{k=1}^n (1-a_k)>\left(1-\sum_{k=1}^{n-1}...

Re: 2012 HSC MX2 Marathon A pretty fast way to prove the "reflective" properties of conics. Let ABC be a triangle. Let X be a point on BC. i) Prove that AB/BX = AC/CX if and only if the line segment AX bisects the angle BAC. ii) Using the above, state and prove the reflective property of the...

Well that's just the standard notation for the function. Has been used for at least a hundred years now. A standard notation is needed because more can be said about the distribution of the primes than just the above asymptotic relation above. Nope, I don't think I ever met him...

Primes are beautiful, right now I am writing an article about the Prime Number theorem. (Which at least involves the greek letter pi I guess...) $Let $\pi(x)$ denote the number of primes less than or equal to $x$. Then\\ $\frac{\pi(x)\log(x)}{x}\rightarrow 1$ as $x\rightarrow \infty$. So...

I think Q1 is incorrect...if for example we set n=2, we get A = 2.073 and a lower bound of 2.89 approximately. (There are only three triangles in this case.) It also seems a little difficult/fiddly to determine whether or not the sequence of triangles "wraps all the way around" the common point...

That assertion follows from the fact that [0,1] is a compact subset of R. This is an immediate consequence of the Heine-Borel Theorem. The below direct argument is essentially a special case of how we can prove the Heine-Borel Theorem for boxes in R^n. Suppose [0,1] has an open covering A of...

Re: 2012 HSC MX1 Marathon Comparison to which geometric series? Here is a cute overkill proof that uni students may find amusing: $If the harmonic series converges to some value $H$, then the alternating harmonic series $L:=1-1/2+1/3-\ldots$ is absolutely convergent. Hence we can rearrange...

Didn't go to many lectures so I'm not sure...I think the tools of complex analysis you have available by the end of the course make it fairly trivial though. (Like Liouville's theorem.) So it probably appears as a tute question.

Re: 2012 HSC MX2 Marathon I would say that the most important unsolved problem in mathematics is beyond syllabus :).

Re: 2012 HSC MX2 Marathon $Let $f(x)=\frac{1}{1+x^2}.$ From the definition of the Riemann integral:\\$\pi/4=\int_0^1\frac{dx}{1+x^2}=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^n f(k/n)\cdot n^{-1}\right)=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^n\frac{n}{n^2+k^2}\right).

Lols, yeah. well tweak the questions notation to suit your working...as long as your notation is consistent it doesn't matter too much. Well meaning there exists a z0 in the disc such that f(z)>=f(z0) for all z in the disc. Think of it as an application of the 2d version of the extreme value...

Typo: 'constant'--->complex in 5. Try it guys, it's not too difficult.

That isn't proving the FTA. That is proving that if the FTA holds, then any complex poly of degree n has n roots counting multiplicity. The FTA is USED in the initial step of that induction... (and it is a perfectly formal way of establishing that fact.)

A lot of my students have asked me how to prove the Fundamental Theorem of Algebra. Most of the proofs I have seen make use of some deep result from complex analysis or topology, however I recently came across a proof that is accessible to Extension Two students. Here is a question I wrote that...

Re: 2012 HSC MX2 Marathon Thats not true as stated. Let (n,k)=(1,1),(2,1). The circle is not tangential to the line...