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Re: HSC 2016 4U Marathon - Advanced Level Yep that's the idea, but to clean up the solution, it should be noted that if we started with x =\cos \theta + i \sin \theta then we possibly will not get an infinite chain if we kept squaring it. But, suppose that x= \cos \theta + i \sin \theta...

Re: HSC 2016 4U Marathon Needed: Only complex numbers \\ $Let$ \ \alpha \ $and$ \ \beta \ $be complex numbers so that:$ \\ \alpha - \beta = ik \ $and$ \ \frac{\alpha}{\beta} = im \ $where$ \ k,m \ $are real$ \\ $i) Show that if$ \ z = \alpha + \beta \ $then$ \ z = \frac{2mk}{m^2 + 1} + i...

Re: HSC 2016 4U Marathon - Advanced Level Something a little easier: \\ $Show that no number of the form$ \ 4k+2 \ $or$ \ 4k+3 \ $for any positive integer$ \ k \ $can be a perfect square$

Re: HSC 2016 4U Marathon - Advanced Level I got that too (with p(x) = 0) but I managed to show that it was the only possible solution

Re: HSC 2016 4U Marathon - Advanced Level This is a good one that I just did: \\ $Find all polynomials with real coefficients that satisfy:$ \ p(x) p(x+1) = p(x^2) I don't know if you consider this difficult enough

Re: HSC 2016 4U Marathon - Advanced Level Right, my bad, instead: \\ A^2 = \frac{1}{16} (a+b+c) p(a+b+c) = \frac{1}{32} p''(0) p\left(\frac{p''(0)}{2} \right) \\ $Since$ \ p \ $is a rational polynomial, so must$ \ p'' \ $and thus$ \ p''(0) \ $is rational$ \\ $So,$ \...

Re: HSC 2016 4U Marathon - Advanced Level \\ 5(3^{n+1} - 2^{n+1}) - 6(3^{n} - 2^n) = (3+2)(3^{n+1} - 2^{n+1}) - 3 \cdot 2 \cdot (3^n - 2^n) \\ \\ = 3^{n+2} - 3 \cdot 2^{n+1} + 2 \cdot 3^{n+1} - 2^{n+2} - 2 \cdot 3^{n+1} + 3 \cdot 2^{n+1} = 3^{n+2} - 2^{n+2} \\ \\ \Rightarrow 5(3^{n+1} -...

Re: HSC 2016 4U Marathon - Advanced Level Yep, well done My solution was pretty much the same (I'll write it out in case people can't read your solution) \\ $Let the altitudes of the triangle, intersecting sides$ \ a, b, c \ $be$ \ \alpha, \beta, \gamma \ $respectively$ \\ $Considering the...

Re: HSC 2016 4U Marathon - Advanced Level Perhaps an interesting excercise: \\ $Prove that the function$ \ f(x) = \frac{2x}{1+x^2} \ $is monotone decreasing for$ \ x > 1 \ $without any recourse to calculus$

Re: MX2 2016 Integration Marathon Not intentional haha That substitution requires a constant in front of it, otherwise we can't simplify it to eliminate the square-root

Re: HSC 2016 MX2 Combinatorics Marathon \\ $For coins$ \ c_1, c_2, \dots , c_n \ $we have probability$ \ c_i \ $returning heads is$ \ p_i \ $for all$ \ i = 1,2, \dots , n \\ $Let the probability that the first$ \ m \ $coins return$ \ k \ $coins be$ \ s(m,k) \\ $Find a recurrence...

Re: HSC 2016 4U Marathon - Advanced Level In the future feel free to post your progress, in this case it was just a computation and the integral is not essential to understanding the problem

Re: MX2 2016 Integration Marathon \\ \int_0^{2\cos \theta} \sqrt{x^2 - 2x \cos \theta + 1} \ $d$x \\ $Take$ \ \theta \ $to be constant$

Re: HSC 2016 4U Marathon - Advanced Level Perhaps something easier (with a bit of guidance), this is possible with 3U knowledge and 4U polynomial knowledge: \\ $Note: one of the altitudes of a triangle, is the distance from one of its vertices to its opposite side$ \\ $A triangle has side...

Re: HSC 2016 4U Marathon - Advanced Level \\ $Place this polygon onto a cartesian plane, let the internal angle between each side be$ \ \theta \\ $So, the angle between lines with gradient$ \ m_k \ $and$ \ m_{k+1} \ $is$ \ \tan \theta = \frac{m_k - m_{k+1}}{1 + m_k m_{k+1}} \\ $So,$ \ m_k...

Re: HSC 2016 4U Marathon - Advanced Level

Re: HSC 2016 4U Marathon - Advanced Level \\ $Let$ \ p + \sqrt{q} \ $be a root of the integer polynomial$ \ P \\ $Consider the polynomial$ \ f(x) = ( x - (p + \sqrt{q}))(x - (p - \sqrt{q})) = (x-p)^2 - q \\ f(x) \ $also has root$ \ p - \sqrt{q} \\ $Divide then$ \ P(x) \ $by$ \ f(x) \...

Re: HSC 2016 4U Marathon - Advanced Level

Re: HSC 2016 4U Marathon \\ $The polynomial$ \ ax^3 + bx^2 + bx + a \ $is such that$ \ a,b \ $are positive and all the roots are real, prove that$ \ b \geq 3a

Re: HSC 2016 4U Marathon - Advanced Level Right, then the next step is to show that any integer polynomial P has to have the same property