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The first blue line is incorrect, for example, 1.5 + 2.5 = 4, 4 is an integer, but the sum of 2 numbers being an integer does not mean those numbers are integers. This is just a guess, but I believe you will need to use the fact that 7 is prime. Otherwise your argument can be...

Re: HSC 2014 4U Marathon - Advanced Level Ah yes my bad k_1, k_2, k_3 \in \{-1,1 \} \\ $Case 1:$ \ \ k_1 = k_2 = k_3 = 1 \ \Rightarrow \ $(dealt with above)$ \\ \\ $Case 2:$ \ \ $2 of$ \ k_1, k_2, k_3 \ $is$ \ -1 \ $w.l.o.g$ \ k_1=k_2 = -1 \ , \ k_3 = 1 \\ \\ (2) \Rightarrow \ (c-a) =...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Find$ \ \ \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{n + ik} \\ \\ $Where$ \ i = \sqrt{-1}

Re: HSC 2014 4U Marathon - Advanced Level \\ $Using the theorem, that for integer polynomial$ \ P \ $and distinct integers$ \ a,b \ \ $then$ \ \ \frac{P(a) - P(b)}{a-b} \ \ $is an integer$ \\ $This is easily provable by showing that$ \\ \\ P(x) = \sum_0^n a_kx^k \ \Rightarrow \ P(a) - P(b) =...

I = \int_0^2 (3x(x-1)+1) \cos((x-1)((x-1)^2+1)) \ dx \\ u = x-1 \\ \Rightarrow I = \int_{-1}^1 (3u(u+1)+1) \cos(u(u^2+1)) \ du \\ \\ u = -z \\ \Rightarrow I = \int_{-1}^1 (-3z(1-z) + 1) \cos(z(z^2+1)) \ dz \\ $Adding side by side$ \\ \therefore \ 2I = \int_{-1}^1 (3x(x+1) + 1 + (-3x)(1-x) +...

Yep its just a Riemann sum, I was just disguising my solution to make it HSC worthy

Re: MX2 Integration Marathon \\ u = -x \\ \Rightarrow \ \int_{-\pi/4}^{\pi/4} \frac{e^x}{1+e^x}\sec^3x \ dx = \int_{-\pi/4}^{\pi/4} \frac{1}{1+e^u}\sec^3 u \ du \\ \Rightarrow \ 2I = \int_{-\pi/4}^{\pi/4} \sec^3 x \ dx \Rightarrow \ \dots \dots

Re: HSC 2014 4U Marathon - Advanced Level Great question! Minor corrections: f(x) = \sin x - \sin(2x) \ \ $is convex on$ \ x \in [0, \pi]

Re: HSC 2014 4U Marathon - Advanced Level Nice! Well done. Here is my solution: (Using cyclical sum notation to shorten writing) \\ \sum_{cyc} \frac{a^2}{(a+b)(a+c)} = \sum_{cyc} \frac{1}{\left(1 + \frac{b}{a} \right) \left(1 + \frac{c}{a} \right)} \\ x = \frac{b}{a} \ , \ y = \frac{c}{b} \...

Re: HSC 2014 4U Marathon - Advanced Level Wait is this an actual theorem? What is it called?

Re: MX2 Integration Marathon \int \tan (x) \cdot \tan (2x) \cdot \tan (3x) \ dx

Re: HSC 2014 4U Marathon - Advanced Level \\ $Show for positive real$ \ a,b,c \\ \\ \frac{a^2}{(a+b)(a+c)} + \frac{b^2}{(b+c)(b+a)} + \frac{c^2}{(c+a)(c+b)} \geq \frac{3}{4}

Re: MX2 Integration Marathon \\x = (u-1)^4 \\ \Rightarrow \ I = \int \frac{4(u-1)^3}{(u-1)^2 u^{10}} \ du = 4\int u^{-9} - u^{-10} \ du = \frac{4}{9u^9} - \frac{1}{2u^{8}} + c \ \ \rightarrow \ u = \sqrt[4]{x} + 1 --------- \\ u = x - \frac{1}{x} \\ \Rightarrow \ J = \int \frac{du}{-u...

Re: MX2 Integration Marathon \int \frac{dx}{(x+\alpha)^{\frac{n+1}{n}} (x+\beta)^{\frac{n-1}{n}}}

I believe non-uniform circular motion is not the HSC syllabus, however: \omega = \frac{d\theta}{dt} \ , \ \phi = \frac{d\omega}{dt} x = r\cos \theta \ \Rightarrow \ \dot{x} = - r\omega \sin \theta \ \Rightarrow \ \ddot{x} = -r\phi \sin \theta - r\omega^2 \cos \theta y = r\sin \theta \...

Re: HSC 2014 4U Marathon \\ $Find a root of the equation$ \ \ (x-2)(x-3)(x-4)(x-5) = 24

Re: HSC 2014 4U Marathon - Advanced Level \\ $For$ \ p_4(x) = \sum_{k=0}^4 \frac{x^k}{k!} \ \ \ \lim_{x \to -\infty} = \infty \ \ $and$ \ \ \lim_{x \to \infty} = \infty \\ \\ $Assume$ \ p_4 \ $has a root$ \ \alpha \\ $Either$ \ \alpha \ $is a double root, or if$ \ \alpha \ $is a root, there is...

Yea I can't see how to resolve that without doing some calculus in disguise

How about this: \frac{3\sin x}{2 + \cos x} < x \\ \Rightarrow \ 2x + (x\cos x - 3\sin x) > 0 \\ \Rightarrow \ 2x + \sqrt{x^2+9} \cos \left(x + \tan^{-1} \left(\frac{3}{x}\right) \right) \\ \\ 2x > \sqrt{x^2+9} \ $for$ \ x > \sqrt{3} \\ \therefore \ 2x + \sqrt{x^2+9} \cos f(x) > 0 \ $for$ \ x >...