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I went through a period of being healthier earlier this year: regular gym, healthy food, less alcohol...sadly I have relapsed a bit in all three departments as of the last month or two.

agreed.

whoops, thought I replied to this. you should be fine, its not that difficult and its the sort of subject that doesn't need many prereqs.

~2AM: 3 mcdoubles 8AM: tuna chunks in olive oil on sourdough dinner: rare steak with mushroom sauce, a salad and some fries.

Haha 3U clusterfucks aren't fun.

Re: HSC 2013 4U Marathon Well you don't need to know how to prove the factorial thing, think about the definition of a prime number...it is convenient that we have found a function f that "checks divisibility". Your answer is quite close, but your p(n) will actually spit out...

curious as well.

Re: HSC 2013 4U Marathon $1. For positive integers $d$ and $m$, evaluate:\\ $f(m,d)=\frac{1}{d}\sum_{j=1}^{d} \cos\left(\frac{2\pi jm}{d}\right).$\\ \\2. Use the above to find a function $g(n)$ defined in terms of MX2 functions such that $g(n)=1$ if $n$ is prime and $g(n)=0$ if $n$ is...

bacon <3.

Re: HSC 2013 4U Marathon yep, cheers.

Re: HSC 2013 4U Marathon $Let $(x_1,x_2,\ldots,x_n)$ be an increasing sequence of real numbers and let $(y_1,y_2,\ldots,y_n)$ be an arbitrary rearrangement thereof. \\Prove that the expression $\sum_{j=1}^n x_jy_j$\\ is maximised if the sequence $y$ is increasing, and minimised if $y$ is...

You seem to have no issue with voicing your own.

Re: HSC 2013 4U Marathon \frac{d}{dt}\left(e^{-Bt}\int_0^t g(s)\, ds\right)=\left(g(t)-B\int_0^t g(s)\,ds\right)e^{-Bt}\leq Ae^{-Bt} by the product rule, fundamental theorem of calculus, and our assumed integral inequality. Replacing t with r and integrating this expression from r=0 to...

Re: HSC 2013 4U Marathon Your f is indeed the unique function which gives us equality for all t >= 0 (*), but I don't know how you are deducing that any g satisfying the integral inequality must be smaller than f everywhere (which is what we are aiming for). (*) Your method of showing this is...

Re: HSC 2013 4U Marathon $ $g(t)$ is a continuous function and $A,B$ are positive constants such that:\\ \\ $g(t)\leq A+B\int_0^t g(s)\, ds $\\ \\ for all $t\geq 0.$\\ \\Prove that $g(t)\leq Ae^{Bt}$\\ \\for all $t\geq 0.$ $

Whoops, typo. a,b should be chosen from {1,2,...,p-1}, and we are proving that these "inverses" exist for any prime p. Ie we are just specialising to the case where m=p is prime. PDF is fixed.

Yeah seconded, if the course hasn't changed much you will cover all necessary PDE theory in the course itself, but it is good to be familiar with the basics beforehand so you don't get bogged down when using them in more complex ways. The courses are both of comparable difficulty (imo)...

Lies your teachers have told you Vol1: There is no "formula" for prime numbers. A roughly MX2 level question attached. PS, quicker ways to derive such formulae certainly exist. As an easier side exercise which involves no modular arithmetic, try to construct a formulae for the n-th prime...

Re: HSC 2013 4U Marathon That's about as good as you can do with mx2 definitions and notions, but it is certainly not rigorous as the object you are proving things about (the Riemann integral) is not rigorously defined. Other things to think about: 1. What is the definition of area itself...