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Re: HSC 2014 4U Marathon \\ $Let the sequence$ \ a_{n+1} = pa_n - a_{n-1} \ $have$ \ a_0 = 1 \ $and$ \ a_1 = p \\ \\ $Show that the polynomial$ \ x^n - a_nx + a_{n-1} \ $is divisible by$ \ x^2-px+1

Re: HSC 2014 4U Marathon - Advanced Level Yep that is correct, that is probably a "hidden" version of mine which is basically: a^5 + b^5 \geq a^2b^2(a+b) By considering the factorisation of a^5+b^5 = (a+b).... And then things become quite clear after

Re: HSC 2014 4U Marathon - Advanced Level Oh wait I forgot to put in the condition a^2+b^2+c^2 = 1 Editing it in now.

Re: HSC 2014 4U Marathon - Advanced Level \\ $If$ \ a,b,c \ $are positive reals so that$ \ a^2 + b^2 + c^2 = 1 \ $, prove$ \\ \\ \frac{a^5+b^5}{ab(a+b)} + \frac{b^5+c^5}{bc(b+c)} + \frac{a^5 + c^5}{ac(a+c)} \geq 3(ab + ac +bc) - 2

Re: HSC 2014 2U Marathon I know this is possible with 2U knowledge, but I think these sort of questions are best left to the higher marathons, most students doing 2U anyway don't really want questions of that difficulty

Re: HSC 2014 4U Marathon - Advanced Level I was expecting an answer that has more knowledge grounded in the 4U course Its not really a fair question, but good solution nonetheless even though I understand very little of it

Re: HSC 2014 4U Marathon \\ $i) Show that if the sum of the digits of a number is divisible by 3, then the number is divisible by 3$ \\ \\ $ii) Show that if the obtained by taking the last two digits of the number on its own, is divisible by 4, then the original number is divisible by 4$ \\ \\...

Re: MX2 Integration Marathon \\ \int_{0}^{\infty} \frac{2x^3 - 3x^2 + 1}{x^6 - 2x^3 + x^2 - 2x + 2}

Re: HSC 2014 2U Marathon \\ \int \frac{dx}{x \ln x^{\ln (\ln x)}} \ dx

Re: HSC 2013 3U Marathon Thread .

Re: HSC 2014 4U Marathon - Advanced Level what

Re: HSC 2013 3U Marathon Thread Yea well done

Re: MX2 Integration Marathon \\ \int \frac{1}{(1-x) \sqrt{1+x^2}} \ dx

Re: HSC 2014 2U Marathon \\ $(i) Show that$ \ x^n - 1 = (x-1)(1+ x + x^2 + \dots + x^{n-1}) \\ \\ $(ii) Show that if a number is in the form$ \ 18^n - 1 \ $for some integer$ \ n \ $then such a number is divisible by$ \ 17 \\ \\ $(iii) Using the result in part (i), explain why$ \...

Re: HSC 2013 3U Marathon Thread \\ $Prove by mathematical induction that the sum of the interior angles of an$ \ n \ $sided polygon is$ \ \pi(n-2)

Re: HSC 2014 4U Marathon \\ $Show that if$ \ x \ $is odd, then$ \ x^2-1 \ $is divisible by$ \ 8

Re: HSC 2014 4U Marathon Sorry that was a 'Find' question, not 'Show that', edited

Re: HSC 2014 4U Marathon \\ $Given$ \ \sin \alpha + \sin \beta = \sqrt{\frac{5}{3}} \ , \ \cos \alpha + \cos \beta = 1 \\ \\ $Find$ \ \cos (\alpha - \beta)

Re: HSC 2014 2U Marathon dafuq

Re: HSC 2013 3U Marathon Thread Yea it is, my initial fact was useless ---------- \\ \lim_{c \to \infty} \int_{-c}^c \frac{1}{e^{x} + e^{-x}} \ dx