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Re: HSC 2014 4U Marathon C_0(x) is supposed to be 1, my bad, editing now And yes it should work for x = 0, (see when n=0)

Re: HSC 2014 4U Marathon - Advanced Level \\ $Given polynomial$ \\ P(x) = \sum_{k=0}^n a_n x^n \ $where$ \ a_k \in \mathbb{R} \ $and$ \ a_k > 0 \ $for$ \ k=0,1,2, \dots, n \\ \\ $Show that for any complex root of$ \ P \ $, that is$ \ \alpha \ $then$ \ |$arg$ \ \alpha| \geq \frac{\pi}{n}

Re: MX2 Integration Marathon \\ \int_0^1 \sin \ln x \ dx

Re: HSC 2014 4U Marathon \\ $Let$ \ C_n(\cos \theta) = \cos n \theta \ $for some polynomial$ \ C_n \ $defined on$ \ -1 \leq x \leq 1 \\ $i) Show that$ \\ \\ C_{n}(x) = \begin{cases} 1, \ n = 0 \\ x, \ n= 1 \\ 2xC_{n-1}(x) - C_{n-2}(x), \ n \geq 2 \end{cases} \\ $ii) Using De Moivere's...

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Re: HSC 2014 4U Marathon \\ $Let$ \ ABC \ $be a triangle with point$ \ X \ $on side$ \ BC $. If$ \ AB = c, AC = b, AX = p, BX = m, $and$ \ XC = n, \ $then, prove$ \ ap^2 +amn = b^2m+c^2n.

Re: HSC 2014 4U Marathon - Advanced Level Its just glorified AM-GM (if I didn't make a mistake) \\ \sum_{i < j} (a_i^2 b_j^2 + a_j^2 b_i^2) \geq \sum_{i <j} 2a_i b_i a_j b_j \ \ $(AM-GM)$ \sum_{k=1}^n a_k^2b_k^2 + \sum_{i < j} (a_i^2 b_j^2 + a_j^2 b_i^2) \geq \sum_{i <j} 2a_i b_i a_j b_j +...

Re: MX2 Integration Marathon Don't forget \int e^{x+e^{x}} \ dx

Re: MX2 Integration Marathon Is there a reason for that specific list of functions to be considered elementary? What if I say Elementary = anything formed recursively (using the operations of +,-,*,/,^ and composition) from: the reals, the monomial x, the 3 trig funcs and their inverses...

Re: MX2 Integration Marathon Is there a rigorous definition of "elementary function"? What makes the gamma function (for example) not elementary, but something like ln() or sin() elementary even though they are both transcendental?

Re: MX2 Integration Marathon \\ \int_0^{\frac{\pi}{4}} \ln (1+\tan x) \ $d$x

Re: HSC 2014 4U Marathon - Advanced Level \\ $Find$ \ \sum_{n = 0}^{\infty} \frac{(-1)^n}{3n+1}

Re: MX2 Integration Marathon Can that be found in terms of elementary functions?

Re: MX2 Integration Marathon Yea its something like that With integration I just try to figure out how to do the integration but I don't really do the computing lol e.g. "oh yea this integral you just do this substitution then alter that then partial fractions and compute"

Re: MX2 Integration Marathon You can do LaTeX by surrounding the tex in and tags, no need to put $ signs like mathstackexchange I would just use this: \\ 1 = (\sin^2 x + \cos^2 x)^2 = \sin^4 x + \cos^4 x + 2\sin^2 x \cos^2 x \\ \\ \Rightarrow \ \frac{1}{\sin^4 x + \cos^4 x} = \frac{1}{1 -...

Re: HSC 2014 4U Marathon - Advanced Level \\ 2 = (a^3+b^3) = (a+b)(a^2-ab+b^2) = (a+b)((a+b)^2 - 3ab) \\ = (a+b)^3 + 3(a+b) (-ab) \geq (a+b)^3 + 3(a+b) \frac{-(a+b)^2}{4} = \frac{1}{4}(a+b)^3 \\ \\ 2 \geq \frac{1}{4}(a+b)^3 \Rightarrow \ 8 \geq (a+b)^3 \Rightarrow \ 2 \geq a+b

Re: HSC 2014 4U Marathon - Advanced Level Nice proof! Here is mine: \\ \frac{(a_1 a_2 \dots a_n)^{1/n} + (b_1 b_2 \dots b_n)^{1/n}}{((a_1+b_1)\dots(a_n + b_n))^{1/n}} \\ \\ = \left(\frac{a_1}{a_1+b_1} \dots \frac{a_n}{a_n+b_n} \right)^{1/n} + \left( \left(1 - \frac{a_1}{a_1+b_1} \right)...

Re: HSC 2014 4U Marathon - Advanced Level I don't think that would work Besides, there's a much more elegant solution!

Re: HSC 2014 4U Marathon - Advanced Level \\ $Let$ \ a_k \ $and$ \ b_k \ $be non-negative for$ \ k=1,2,\dots, n \\ \\ $Prove that$ \\\\ (a_1 \cdot a_2 \dots a_n)^{\frac{1}{n}} + (b_1 \cdot b_2 \dots b_n)^{\frac{1}{n}} \leq \left((a_1 + b_1) \cdot (a_2+b_2) \dots (a_n + b_n)...

Re: HSC 2014 4U Marathon \\ $4 points with$ \ x$-coordinates$ \ x_1, x_2, x_3, x_4 \ $lie on the hyperbola$ \ xy = 1 \ $show that if these 4 points form a cyclic quadrilateral, then$ \ x_1x_2x_3x_4 = 1