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Re: HSC 2013 4U Marathon Moving on from part (i) LHS = \frac{(x+i)^{2n}}{(\cot^2 \theta + 1)^n} = \sin^{2n} \theta (x+i)^{2n} = \cos 2n\theta + i\sin 2n\theta Equate real parts of both sides, also make the substitution: y=x^2 \frac{\cos(2n\theta)}{\sin^{2n} \theta} = y^n - \binom{2n}{2}...

Re: HSC 2013 4U Marathon Give it a bit of time, I'm working on it right now :P

Re: HSC 2013 4U Marathon \int \frac{1-x}{1-\sqrt{x}} \ dx = \int \frac{(1-\sqrt{x})(1+\sqrt{x})}{1-\sqrt{x}} = \int 1+\sqrt{x} \ dx = x + \frac{2}{3}x^{\frac{3}{2}} + c ========== \int \frac{dx}{e^x + e^{-x}}

Re: HSC 2013 4U Marathon From the tangent-secant theorem: PQ^2 = PA \cdot PB PQ^2 = (PM - AM)(PM+BM) $Since$ \ \ AM=BM PQ^2 = PM^2 -AM^2 AM^2 \geq 0 \therefore PM^2 \geq PQ^2 \ \ \Rightarrow PM \geq PQ ii) [tex]$In order for the inequality to establish equality (i.e. when PM =...

Re: HSC 2013 4U Marathon I'll give someone else a go for this one, good question =)

Re: HSC 2013 4U Marathon (Another early IMO question). $Solve the equation$ \cos^2 x + \cos^2(2x) + \cos^2(3x) = 1

Re: HSC 2013 4U Marathon Wow...... Very intelligent solution. Mine was simplifying the number where it must satisfy: 13n = 2 x 10^{m} - 2 for some integer m, and then I just computed a couple of values until I arrived at what I needed. Though your method completely trumps mine haha

Re: HSC 2013 4U Marathon Yep haha, nice work.

Re: HSC 2013 4U Marathon Hmm is there perhaps a really elegant solution here? It is an early IMO question and I was able to solve it with a bit of algebra and substitution.

Re: HSC 2013 4U Marathon $Find the smallest natural number$ \ \ n $which satisfies the following properties$ $1) It has 6 as its last digit$ $2) If the last digit 6 is erased and placed in front of the remaining digits$ $ the resulting number is four times as large as the original...

Re: HSC 2013 4U Marathon $A cyclic quadrilateral has the side lengths$ \ \ a, b, c, d $Prove that the area of the cyclic quadrilateral is given by$ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} $Where$ \ \ s=\frac{1}{2} (a+b+c+d)

Re: HSC 2013 3U Marathon Thread Lol don't worry about that question, I stuffed it up.

Re: HSC 2013 4U Marathon You may be correct, I did a the substitution x=sin^2 u which got me to a trig recurrence. So maybe I made a mistake. But I'm not sure I don't have the answer with me.

Re: HSC 2013 3U Marathon Thread $By utilising an appropriate substitution, Find$ \int \sqrt{x -2x\sin^2x + x\sin^4 x} \ dx

Re: HSC 2013 4U Marathon $If$ \ \ I_n = \int_0^1 x^n \sqrt{x(1-x)} \ dx $Prove that$ I_n = \frac{4n+2}{4n+5} I_{n-1}

Its Omed62, just ignore him.

Re: HSC 2013 4U Marathon $If$ \ \ y=\cos(n \cdot \cos^{-1}x) $i) Prove that$ y= \frac{1}{2} ((x+\sqrt{1-x^2})^n + (x-\sqrt{1-x^2})^n) $ii) Prove that if$ \ \ n \ \ $is an odd number, it can be expressed as a polynomial$ y = a_k x^{2k+1} + a_{k-1}x^{2k-1} + \dots + a_1 x^3 + a_0 x...

Re: HSC 2013 4U Marathon \alpha + \beta = \frac{\pi}{2} - \gamma \sin(\alpha + \beta) = \cos(\gamma) \therefore \ \ \sin(\alpha + \beta) \sin(\alpha + \gamma) \sin(\beta + \gamma) = \cos \alpha \cos \beta \cos \gamma=RHS RHS =\sin(\beta + \gamma) (\frac{1}{2}(\cos(\alpha + \beta -\alpha...

Re: HSC 2013 4U Marathon yep, by AM-GM

Re: HSC 2013 4U Marathon Yeah I imagined the whole thing completely wrong, I better go to sleep.