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Re: Extracurricular Integration Marathon Still isn't taught too much really, but it's pretty useful! That's why I wrote that question like last month in the undergrad marathon about differentiating things of the form \int_{a(t)}^{b(t)}f(x,t)\, dx w.r.t. t, as that is just a strengthened form...

Re: Extracurricular Integration Marathon 3. \lim_{T\rightarrow \infty}\int_{-T}^T \frac{\sin(x)}{x}\, dx 4. \lim_{T\rightarrow \infty}\int_{-T}^T \sin(x^2)\, dx

Re: Extracurricular Integration Marathon Yep :).

Re: Extracurricular Integration Marathon Which follows from differentiating under the integral (Feynman's favourite integration trick): I'(a)=\int_0^1 x^a=\frac{1}{a+1}, I(0)=0\\ \\ \Rightarrow I(a)=I(0)+\int_0^a I'(s)\, ds = \int_0^a \frac{ds}{s+1}.

Re: HSC 2016 4U Marathon - Advanced Level By MX2 I meant doable by an MX2 student i.e. in the scope of this thread. I didn't mean that you would only be capable of doing it if you knew the MX2 course.

Re: HSC 2016 4U Marathon - Advanced Level That is better, and more or less what I wanted people to say for 2. One thing though: Monotonicity isn't so essential here if you have already calculated R(n) explicitly. What is important is just that each R(n) is smaller than 1/4*pi. (*)...

Re: HSC 2016 4U Marathon - Advanced Level Haha yeah, sorry about that. The start was just boredom / curiosity at how some of the more sexist people on this forum would react (as there are/were plenty of them). After a while people got to know the account though so I began to mostly use that...

Re: HSC 2016 4U Marathon - Advanced Level Err, no. Although I am not sure if you are just joking by talking about infiniti-eth ratios.

Re: HSC 2016 4U Marathon - Advanced Level Nothing too fancy, just the standard tool for optimising functions w.r.t constraints. (e.g. Maximise e^(ye^x) subject to the constraint x^2+y^2=1). Looking at stationary points and doing the higher dimensional analogues of the "derivative tests" to...

Re: HSC 2016 4U Marathon - Advanced Level Indeed there is. Consider an n-gon that is optimal w.r.t. the n-gon isoperimetric inequality. Now construct an n+1 gon by severing two of the adjacent vertices of the n-gon and connecting them both to a new point Q, chosen in a way that avoids...

Re: HSC 2016 4U Marathon - Advanced Level Really? Could you possibly link me one of the ones corresponding to the second if you can find it? Genuinely interested, as I haven't encountered these particular pathological curves before. The first would also interest me depending on what you...

Re: HSC 2016 4U Marathon - Advanced Level Yeah, after a while I just got lazier and lazier hahaha.

Re: HSC 2016 4U Marathon - Advanced Level Oh lols, gave away my alt. Ah well, it's done now. :p

Re: HSC 2016 4U Marathon - Advanced Level Also, by topological solutions of the isoperimetric inequality, do you mean: -a proof by largely topological means that A/P^2 cannot exceed 1/4*pi, with equality only at circles. (quantifying over a set of curves of some regularity class.) -a...

Re: HSC 2016 4U Marathon - Advanced Level I know. As I said, the isoperimetric problem (ie showing that the optimal area-enclosing curves amongst curves of a certain regularity are precisely the circles, and the optimal value is 1/4pi) is not suitable as a high school problem. (It would be...

Re: 2015 permutation X2 marathon Note the words "in general", I was just giving a high brow overview of how this stuff generalises. You do not need double integration for this particular question, as my single integral expression a few posts above shows. Idk about 2U, but such examples of...

Re: 2015 permutation X2 marathon Basically, but areas is a restrictive way to think about it imo its is more of a ratio of sets in your probability space (space of outcomes). In this case your probability space is a product measure space so in general we would need a double integral (weighted...

Re: 2015 permutation X2 marathon I assume braintic wants the answer in case form, so I will let someone else do that :).

Re: 2015 permutation X2 marathon \mathcal{P}=\frac{1}{2ab}\int_{-a}^a \min(\frac{p^2}{4},b)\, dp

Re: 2015 permutation X2 marathon Sure, P(A > B)=P(log(A) > log(B)) for positive random variables, this is rather trivial. What is the point of this log transformation for braintics original question though?