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Re: HSC 2015 4U Marathon Good question and nice solution! I don't think you have a full solution though, as your "k" is only "n/2" (what if n is odd?), instead of it being (n-1) So I would square that result you have, because \cos^2 \frac{\pi}{n} = \cos \left(\frac{\pi}{n}\right) \cdot \cos...

Re: HSC 2015 4U Marathon Check your solution by substituting some outlandish values of n (such as n=4, 7, 11 etc.)

Re: HSC 2015 4U Marathon Is that last fraction \cos \frac{2k\pi+\pi}{n} ?

Re: HSC 2015 4U Marathon FrankXie's method is good but another way that I can see is to: \\ (x^3+2x^2+3x+3)(-x^3+2x^2-3x+3) \\ = ((2x^2+3) + (x^3+3x))((2x^2+3) - (x^3+3x)) \\ = (2x^2+3)^2 - (x^3+3x)^2 = \dots

Re: HSC 2015 4U Marathon Easier way: p(x) = x^3 + 2x^2 + 3x + 3 p(-x) = -x^3 + 2x^2 - 3x + 3 p(x) \ $has roots$ \ \alpha, \beta , \gamma p(-x) \ $has roots$ \ -\alpha, -\beta, -\gamma \\ $Therefore$ \ p(x) \cdot p(-x) = q(x) \ $has all these roots$

Re: HSC 2015 4U Marathon Summarised: - Divide top and bottom by x^2 \int \frac{1- 1/x^2}{(x+1/x)\sqrt{x^2+1/x^2}} \ dx - Make the (x^2 + 1/x^2) = (x+1/x)^2 - 2 in the ssquare root - Substitute u = x + 1/x - After that substitute z = 1/u And it becomes a standard integral after that

Re: HSC 2015 4U Marathon http://i.imgur.com/ZiWjEqZ.png

http://i.imgur.com/9CYSHXz.png

You need a dunce cap like the one above

Re: HSC 2015 4U Marathon - Advanced Level I used Re\left(\sum_{n=1}^{\infty} (\cos x + i\sin x)^n \right) and did stuff with it, I think you can see what I attempted, did I do anything wrong since I'm getting definite finite answers?

Re: HSC 2015 4U Marathon - Advanced Level Yes my most recent post with a new question won't load the LaTeX for me either, but earlier posts work just fine text

Re: HSC 2015 4U Marathon - Advanced Level Here is something I made: \\ $Find$ \ \sum_{n=1}^{\infty} \cos \left(\frac{1}{2}n \right) \cos \left(\frac{3}{2}n \right)

Re: HSC 2015 4U Marathon - Advanced Level Oh right that's a typo, the inequality is flipped after that line

Re: MX2 2015 Integration Marathon \\ \int \frac{\sin (3x)}{\sin x} \ dx

Re: HSC 2015 4U Marathon - Advanced Level I found this harder than the first inequality, probably because the answer doesn't require much algebra! \\ $There are two cases to consider$ \\ $Case 1:$ \ \sigma (k) \leq k \ \Rightarrow \ x_k y_{\sigma (k)} \leq x_k y_k \\ \\ $Case 2:$ \ \sigma (k)...

Re: HSC 2015 4U Marathon - Advanced Level \\ x^2 + y^2 \geq 2xy \\ \\ x^2 - xy + y^2 \geq xy \\ \\ \therefore \ x^3 + y^3 = (x+y)(x^2 -xy +y^2) \geq xy(x+y) \ $for$ \ x,y \ $positive$ \\\\ \therefore \ x^3 + y^3 + xyz \geq xy(x+y+z) \\ \\ \therefore \ \frac{xyz}{x^3 + y^3 + xyz} \leq...

You lack basic comprehension of rational arguments, so I guess it is better for you to just leave without embarrassing yourself further

I'm not saying you have to accept anything, I'm saying you need to know what you're talking about. If I wanted to debate economic policy with someone, and I was ignorant of anything to do with economics, you'd rightly call me and fool and ignore me. Now think of the courtesy I'm extending to you...

Yeah, God is eternal, it is not a conundrum, the fact that you are bringing it up shows you have zero understanding about anything remotely theological So since God is eternal, it doesn't make sense to ask "who caused God if God came into being" since God didn't come into being. Obviously that...

God is not a person God is not a big old man in the sky with a white beard This needs to be understood, because quite a lot of people are really ignorant with regards to this fact.