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An interesting extension is: $Find: $\\ \prod_{n=1}^\infty \left(1-\frac{P(n)}{Q(n)}\right)\\ \\ $where $P(n)<Q(n)$ are polynomials that are positive at positive integers n, and $\deg(P)=\deg(Q)-1.$\\ \\ What can you say about the limit if $\deg(Q)>\deg(P)+1$? $

Re: HSC 2017 MX2 Marathon Suppose you have n points on a circle such that no three distinct chords coincide at any single point. How many regions do the nC2 chords divide the interior of the circle into? (You don't have to provide rigorous proof for this question if you can guess the answer...

Am glad they are recorded these days. They were not when I did my undergrad, and 8 was just too early for me with my travel time to uni. Ended up going to verrrrrry few.

\left(\int_a^b f(x)\, dx\right)/ \left(\int_a^b g(x)\, dx\right)\neq \int_a^b \frac{f(x)}{g(x)}\, dx in general. Eg let a=0, f(x)=x^2, g(x)=x.

Yes. HSC Physics is a total joke.

In high school you don't define the exponential function per se, you define the real number e as the unique real number a such that a^x has slope 1 at the origin, and then you work with e^x from there. This is a terribly nonrigorous, because high school mathematics does not rigorously define the...

The answer here mildly depends on: a) your definition of the exponential function. b) whether theta ranges over R or C. (and hence whether the differentiation refers to differentiation with respect to a complex variable or a real variable) c) which facts along the lines of the chain rule you...

Re: First Year Linear Algebra Marathon Of course it is important that a student knows why some of these Markov chain facts are true, so here is a followup q: Suppose you have a random sequence on a finite set A (your state space). The "randomness" is precisely that if the n-th term in this...

Re: First Year Uni Calculus Marathon Here is one instructive way to do it (probably not the fastest way): First choose more natural coordinates, u_n=x_n-1, v_n=y_n-2 centred at the fixed point. Our recurrence then transforms to: \begin{pmatrix}u_{n+1}\\...

Re: First Year Linear Algebra Marathon Let p(n) be points in R^m defined recursively by p(n+1)=Ap(n), where A is a real mxm matrix. 1. Suppose that the eigenvalues of A all have modulus less than 1. What can you say about the limiting behaviour of the sequence p(n), for typical p(0)? 2...

Re: HSC 2017 4U Marathon That's just hazy high school science definitions. There are various precise definitions of lines/curves of best fit (regression analysis, interpolating polynomials etc), and whenever such an object is named a line it is definitely a line in the usual sense of the word...

Re: HSC 2017 4U Marathon :p

Re: First Year Linear Algebra Marathon First year courses have different content at different universities, and lecturers who write exams for these courses are way less constrained by things like syllabi than the people who write HSC exams. Think of this thread as a place to post questions...

Re: First Year Uni Calculus Marathon A glimpse of one of the first subtleties of measure theory: Consider the set of points on the unit circle C. Suppose we want a way of measuring the size/length of arbitrary subsets of C. I.e. we want a function m that maps arbitrary subsets of C to...

$I would probably write something like the following (to avoid having to use cases for the expressions for arg in various parts of the complex plane):\\ \\ $|z_1+z_2|=|z_1|+|z_2|\\ \\ \Rightarrow (z_1+z_2)(\bar{z}_1+\bar{z}_2)=(|z_1|+|z_2|)^2\\ \\ \Rightarrow z_1\bar{z}_2+\bar{z}_1 z_2 =...

Care to provide an example where this can occur?

Re: First Year Uni Calculus Marathon a_2=1/a_1=1. a_3=2/a_2=2. a_4=3/a_3=3/2. ...

Re: HSC 2017 4U Marathon Note that the above yields an end answer of something like \left(a+\frac{nd}{2}\right)\binom{2n}{n}. Busy prepping for a meeting right now but paradoxica can confirm if that is what he was looking for.

Re: HSC 2017 4U Marathon For odd n, I don't think your S_2 is an integer. I think you don't need that subtraction by 1. Also note that you can evaluate the sums C(n,k)^2 and kC(n,k)^2 combinatorially. (This takes less effort than other methods of computation imo.) These sums amount to: A...

Re: First Year Linear Algebra Marathon I understood what you were asking, but the arbitrary field proofs of CH that I know of are rather short, so it seemed an odd detour to go via the complex special case. In any case, I think you can make your idea rigorous but it takes a bit more care and...