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Re: HSC 2015 4U Marathon The first argument doesn't work, just because the sum of all of them is zero doesn't mean all of them are zero (for instance 1 + (-1) +2 + (-2) = 0 but they aren't all zero). Try induction! (Also yea the property "S_m(n) has roots n=0 and n=-1" is only true for m >=...

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Re: HSC 2015 4U Marathon \\ $Let$ \ S_m(n) = 1^m +2^m + \dots +n^m \ $for integers$ \ m \geq 0 \\ $i) By considering the expansion of$ \ (k+1)^{m+1} - k^{m+1} \ $show that$ \\ \\ 1 + \sum_{k=0}^m S_k(n) \binom{m+1}{k} = (n+1)^{m+1} \\ $ii) Hence or otherwise show that$ \ S_m(n) \ $ (with$ \...

Re: HSC 2015 3U Marathon \\ $i) By writing$ \ (1+x)^2 = 1 + x(2+x) \ $show that$ \ \binom{4n}{2n} \ $is equal to the coefficient of$ \ x^{2n} \ $in the expression$ \ \sum_{k=0}^{2n} \binom{2n}{k}x^k(2+x)^k \\ \\ $ii) Hence show that$ \\\\ \binom{4n}{2n} = \binom{2n}{n}\binom{n}{n} +...

Try posting some of the questions on the marathon and don't mention that it's from the CSSA, then wait for someone to answer it

Assumptions: (1) - You need to assume that the options "full-time" and "part-time" are exhaustive in so far as there is no excluded middle (2) - You must also assume that it is possible that there is more than one office (3) - There exists employees in the NY office (*) Without these...

Re: MX2 2015 Integration Marathon \\ $Let$ \ u_n = \frac{n!}{n^{n+1/2}e^{-n}} \\ $Using the trapezium rule we, and that all components of$ \ u_n \ $are positive, then$ \ 0 < u_n < e, \ (1) \\ \frac{u_{n+1}}{u_n} = \frac{e}{\left(1 + \frac{1}{n}\right)^{n+1/2}} \\ $Let$ \ f(x) = \ln...

Re: HSC 2015 4U Marathon

Depends on the school From my knowledge some are stiffer on this than others (for instance I know that particular selective schools will not allow you to take days off before the trials for no real reason because this is seen as an unfair advantage) To be honest you will probably still need...

Study Trials can have quite the effect on your rankings, however make sure you do not bear too much anxiety over trials since you can still improve in HSC exams if you do not do too well Regardless trials are a great opportunity to set you up to do good in the HSC exams - do not forsake this...

I wish the best for all of you in 2016

Re: HSC 2015 4U Marathon - Advanced Level I tried that but couldn't find a conclusive proof for it. The proof is easy if one knows (a latter conjecture that I knew I had to prove but could not) that there cannot exist a line that goes through all intersection points of degree 2 and goes...

Re: HSC 2015 4U Marathon - Advanced Level the line wouldn't come up for some reason, edited

Re: HSC 2015 4U Marathon - Advanced Level I've tried for too long at this idea and I can't get to a proof of it New question \\ $Suppose that there are some lines in a plane, no pair of which are parallel, and they do not all intersect at a single point. Prove or disprove the conjecture, 'in...

I like this one :D

And still, after all this time, The sun never says to the earth, "You owe Me." Look what happens with A love like that, It lights the Whole Sky. ― Hāfez

I don't think so at all, the first proposition is a direct read off of your posts and are propositions you must hold in any case, you must hold that the Son is God, you must hold that the Father is God, and you must hold that they are distinct Being. For example the Son was supposedly crucified...

THE ROAD NOT TAKEN - Robert Frost Two roads diverged in a yellow wood, And sorry I could not travel both And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth; Then took the other, as just as fair, And having perhaps the better claim...

THE PROGRESS OF MAN - Rumi First he appeared in the realm inanimate; Thence came into the world of plants and lived The plant-life many a year, nor called to mind What he had been; then took the onward way To animal existence, and once more Remembers naught of what life vegetive...

Sure that's the orthodox creed, but the propositions that I've layed out for you are extractions from the creed. Do you deny the first second or third proposition?