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Re: HSC 2018 MX2 Marathon \\ $i) A point $ P $ moves on the rectangular hyperbola $ x^2 - y^2 = a^2. $ Let $ M $ be the point on the tangent to the hyperbola at $ P $ be the point closest to the origin. \\\\Show that this curve is given by the locus $ \ (x^2 + y^2)^2 = a^2(x^2 - y^2). \\\\...

\\ $Almost all real numbers have a (roughly; aka in the limit) equal proportion of 0s and 1s in their binary expansion.$ \\ $To see this, note that if$ \ \{B_n\}_{n\geq 1} \ $are a sequence of iid Bernoulli(0.5) trials, then the random variable$ \ Z_{0.5} = \sum_{n=1}^{\infty}\frac{B_n}{2^n}...

Re: HSC 2018 MX2 Marathon \\ $Referring to the diagram above, where the point$ \ P \ $is the intersection of curves$ \ y =x, \ y = \cos x, $ which region is larger in area, $ A_1 $ or $ A_2 $? Prove your answer without any use of a calculator (except for very basic facts like that $ \pi > 3...

Anyone have a copy?

Re: HSC 2018 MX2 Marathon I made a typo in writing the first question, it should be fixed now. Your second answer however has a 'z' there but this is not a proper cartesian equation for the locus, you want only 'x's and 'y's

Re: HSC 2017-2018 MX2 Marathon ADVANCED \\ $Let$ \ u_n \ $be a sequence defined by, $ \ u_0 = u_1 = u_2 = 1 \ $and, $ \\ u_n u_{n+3} - u_{n+1} u_{n+2} = n! \\\\ $Show that$ \ u_n \ $ is an integer for all $ n \geq 0

Re: HSC 2018 MX2 Marathon \\ $Consider the function in the complex plane, $ \ f(z) = z + i\text{Im}(z). \\\\ $i) Find a locus in the complex plane, where for every$ \ z \ $that lies on that locus, then$ \ |f(z)| = 1 \\\\ $ii) Find the locus in the complex plane of$ \ f(z) \ $for all$ \ |z| = 1...

Re: HSC 2017 MX2 Marathon \\ $Suppose you are analysing the decay of particles from a radioactive source, suppose you discover that the probability that the source emits $ \ k \ $ particles from your source in an hour is$ \\\\ p_k = \frac{e^{-\lambda} \lambda^k}{k!}, \ k \geq 0 \\\\ $Where$ \...

Re: HSC 2017 MX2 Marathon This may not be in the spirit of the question, but if you had something different in mind with a more geometric proof, I'd like to see it. Let O_j be the centre of circle \Gamma_j . Consider the kite MPO_j Q_j , and apply the cosine rule to the side PQ_j from both...

Re: HSC 2017 MX2 Marathon Part (b) is a little long and requires good understanding of the problem, so keep that in mind \\ $Consider the unit circle$ \ x^2 + y^2 =1 \ $and the parameterisation$ \ C_t\left(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right) \ $and denote the point$ \ O(-1,0) \\...

Re: HSC 2016 4U Marathon - Advanced Level This defeats the purpose of the question, if someone wants to do it this way, they'll need to prove the facts that they'd need to, to use it, along the way.

Re: HSC 2016 4U Marathon - Advanced Level Should be simple, but it does not fit in the regular marathon: \\ $Let$ \ a,b,c \ $be integers such that$ \ a\sqrt{2} + b\sqrt{3} + c = 0$, prove that$ \ a = b = c = 0

Re: HSC 2016 4U Marathon \\ P(a \cos \alpha, b\sin \alpha) \ $and$ \ Q(a \cos \beta, b \sin \beta) \ $are points that lie on the ellipse$ \\\\ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \\\\ $It is given that$ \ \alpha + \beta = \frac{\pi}{2} \\\\ $i) Find the coordinates of the midpoint$ \ M \ $of...

Re: HSC 2016 4U Marathon Bump, and another question (suitable for this thread): \\ $Show that$ \ 2^{n+2} + 3^{2n+1} \ $is divisible by$ \ 7 \ $for all integers$ \ n \geq 1

Re: HSC 2016 4U Marathon - Advanced Level \\ $Extend our set$ \ A \ $as follows with the additional rule:$ \\\\ $- If$ \ \alpha \ $is a sentence in$ \ A \ $then$ \ (f\alpha) \ $(denoting some function$ \ f $) is in$ \ A \\ $If$ \ a \ $is a sentence in$ \ A \ $let$ \ l(a) \ $be the number of...

Re: HSC 2016 4U Marathon - Advanced Level \\ $Consider the set of well-formed arithmetic sentences$ \ A \ $defined inductively as follows$ \\\\ $1. Any variable symbol denoting a variable is in$ \ A \\ $2. If$ \ \alpha \ $and$ \ \beta \ $are both sentences in$ \ A \ $then$ \ (\alpha + \beta)...

Re: HSC 2016 4U Marathon See my expanded question for a more 4U approach

Re: HSC 2016 4U Marathon Mostly just algebra and following instructions, it is a fine exercise \\ $Let$ \ I_n = \int_{0}^{ \frac{\pi}{2} } \sin^{2n+1}(x) \ $d$x \ $for integers$ \ n \geq 0 \\\\ $i) Show that$ \ I_n = \frac{2n}{2n+1}I_{n-1} \ $using integration by parts$ \\\\ $ii) Hence show...

Re: HSC 2016 4U Marathon You've got it. I should of probably of posted it in the advanced thread, having a separate logic thread is going to separate everything too much.

Re: HSC 2016 4U Marathon - Advanced Level \\ $1. Considering a polynomial of degree$ \ d \ $then$ \ f(n) = \sum_{i=0}^d a_i n^i, (\Delta f)(n) = \sum_{i=0}^d a_i ((n+1)^i - n^i) = \sum_{i=0}^d a_i(\sum_{j=0}^{i-1} \binom{i}{j} n^j) \\\\ $The inner sum is a polynomial of degree$ \ i-1 \ $and...