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Re: HSC 2014 4U Marathon - Advanced Level \\ $Show that only the constant function$ \ f \ $satisfies$ \ f(x) = f(x^2) \ $for all real$ \ x

\\ $By partially differentiating, we are simply differentiating the function with respect to the variable treating all else as constant$ \\ \\ $So$ \ \frac{\partial F}{\partial x} = \frac{d}{dx} f(x^2-y) = 2x f'(x^2-y) \\ \\ \frac{\partial F}{\partial y} = -f'(x^2-y) \\ $It is clear from these...

The br stuff appeared since you finished the line with pressing enter Is the above what you meant to ask? What does little f mean?

Re: HSC 2013 3U Marathon Thread \\ $Find the domain and range of the following functions$ \\ $i)$ \ f(x) = \sin^{-1}(x) + \sin^{-1} \left( \frac{1}{x} \right) \\ \\ $ii)$ \ g(x) = \tan^{-1}(x) + \tan^{-1} \left( \frac{1}{x} \right)

Re: MX2 Integration Marathon Just thought of this \\ u = \frac{1}{x} \ \Rightarrow \ -\int_1^t \frac{du}{\sqrt{2 - (u-1)^2}} Which is just sine inverse =)

Re: MX2 Integration Marathon (x+1) = \sqrt{2} \sec u \\ $The integral is transformed to$ \ \int_{\pi/4}^{\pi/2} \frac{\sec u}{\sqrt{2} \sec u - 1} \ du = \int_{\pi /4}^{\pi/2} \frac{\sqrt{2} + \cos u}{1 + \sin^2 u} \ du \\ \\ = \sqrt{2}\int_{\pi /4}^{\pi /2} \frac{\sec^2 u}{1 + 2\tan^2u} +...

Re: HSC 2014 4U Marathon - Advanced Level Here was my solution: \\ $It is needed to prove$ \ \frac{a^2+b^2}{2\sqrt{a^2+b^2 + 2ab\cos C}} \leq R \\ \\ $This translates to$ \\ \\ 4R^2(a^2+b^2+2ab\cos C) \geq (a^2+b^2)^2 \\ \\ $By dividing the triangle into 3 by its circumcentre, and looking at...

Re: HSC 2014 4U Marathon - Advanced Level Don't reveal the substitution, that takes all the fun away :/

Re: HSC 2014 4U Marathon - Advanced Level Another one, I can post my solution to the last question if requested: \\ $Find, with proof, all polynomials with real coefficients$ \ P(x) \ $such that$ \\ P(P(x)) = (P(x))^k

In reply to your PM (your box is full): InshaAllah Thanks man much appreciated!

Re: HSC 2014 4U Marathon - Advanced Level Wow much shorter than what I had in mind, using cosine rules and manipulations, and clever last inequality Are you an olympiad student?

Re: HSC 2014 4U Marathon Do you mean arg(z-(3+4i))-arg(z-(2-2i))=\frac{\pi}{3} ? --------- \\ $Let$ \ a = \sqrt[n]{n} \\ \\ $Consider$ \ \underbrace{a^{a^{\dots ^{a}}}}_{n} \ \ $and$ \ n \\ \\ $Which is bigger?$

Re: HSC 2014 4U Marathon - Advanced Level Here is a good one: \\ $The circumradius of a triangle is the radius of the circle that circumscribes the triangle$ \\ \\ $Let the cirumradius of a triangle with sides$ \ a,b,c \ $be$ \ R \\ \\ $Show that$ \\ \\ R \geq \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}

Re: HSC 2014 4U Marathon - Advanced Level Yep exactly my solution

Re: HSC 2013 3U Marathon Thread \\ $Using the Binomial Theorem, find all real roots of the polynomial$ \\\\ x^3 + 4x^2 + 6x + 4

Re: HSC 2014 4U Marathon - Advanced Level Again not too hard: \\ $Given$ \ a,b,c > 0 \ $prove that$ \\ \\ \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \geq \frac{3}{2}

Re: HSC 2014 4U Marathon \\ $If$ \ x,y,z \ $are non-negative real numbers, show that$ \\ \\ (x^2+1)(y^2+1)(z^2+1) \geq (x+y+z- xyz)^2

Re: HSC 2014 4U Marathon - Advanced Level Not that hard compared to other problems in this thread but I think it belongs here more than in the other marathon: $Prove$ \ \binom{n}{k} \leq \frac{n^n}{k^k (n-k)^{(n-k)}}

Re: HSC 2014 4U Marathon Yea I did it via angles as well

Re: HSC 2014 4U Marathon - Advanced Level Yes But if this was presented as a HSC question it would be split up into at the very least 3 parts