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Re: HSC 2016 4U Marathon - Advanced Level The following question is as much a test of choosing good notation in possibly unfamiliar situations as it is of your problem solving abilities. $A polynomial in multiple variables $P(x_1,\ldots,x_n)$ is a formal expression formed by combining sums...

Re: MX2 2016 Integration Marathon Or more generally the inverse functions of ANYTHING we already know how to integrate :).

Re: HSC 2016 4U Marathon It is literally a matter of definition. My use of the word "convex" is exactly that in https://en.wikipedia.org/wiki/Convex_function (as well as most textbooks I have ever seen it in). Using this terminology, x is a convex function, and log(x) is a strictly concave...

Re: HSC 2016 4U Marathon A followup question based on the second derivative test for convexity: Given that f''(x)>0 for all a =< x =< b, show that f is convex on the interval [a,b]. That is, show that f(tx+(1-t)y) =< tf(x)+(1-t)f(y) for all x and y in [a,b], and all t in [0,1]. (This result...

Re: HSC 2016 4U Marathon i) Let g=e^f g(tx+(1-t)y) =< e^(tf(x)+(1-t)f(y)) (convexity of f) = (e^f(x))^t(e^f(y))^(1-t) = g(x)^tg(y)^(1-t) =< tg(x)+(1-t)g(t) (weighted AM-GM in two variables *) for all 0 =< t =< 1. ii) f(x)=log(x) is a counterexample, it is strictly concave as it's...

Re: HSC 2016 MX2 Combinatorics Marathon Exactly, well done :).

Re: HSC 2016 MX2 Combinatorics Marathon Not quite. 1/p is indeed correct for k=1. However I think you have misinterpreted the problem judging by your induction. A sequence terminates after k consecutive heads, rather than just k heads in total.

Re: HSC 2016 MX2 Combinatorics Marathon Also note that we are taking the average of a quantity that can take infinitely many positive values (the process might take an arbitrarily large number of flips to terminate). This is done exactly the same way as averaging something that only takes...

Re: HSC 2016 MX2 Combinatorics Marathon A potentially biased coin produces a H when flipped with probability p. If you flip this coin repeatedly until you get k H's in a row (k being an arbitrary positive integer), what is the average number of flips you will have to do in total? Your answer...

They are indeed, and I was not providing a formal argument there but more a vague justification for my personal point of view (I do not think a convincing argument could be made either way on this matter). Many people might think of real numbers as more tangible and observable in our world than...

I firmly stand on the formalist side of the fence mostly. Although our choice of axioms for our formal logical systems are motivated by patterns witnessed in science/nature/human experience. In that sense many of the formal mathematical objects (eg natural numbers) make a lot of intuitive...

Re: MX2 2016 Integration Marathon Your integral should be \int_0^1 \frac{x^4(1-x)^4}{1+x^2}\, dx. Will leave this for a current student to do though.

Re: MX2 2016 Integration Marathon I=\frac{1}{2^6}\int \sin^6(2x)\, dx\\ \\ = -\frac{1}{2^{12}} \int(\textrm{cis}(2x)-\textrm{cis}(-2x))^6 \, dx\\ \\=-\frac{1}{2^{12}}\int (\textrm{cis}(12x)+\textrm{cis}(-12x))-6(\textrm{cis}(8x)+\textrm{cis}(-8x))+15( \textrm{cis}(4x) +\textrm{cis}(-4x))-20 \...

Re: MX2 2016 Integration Marathon That example doesn't really have anything to do with periodicity. It so happens that the functions sin(x) and cos(x) are periodic, but this property is pretty irrelevant to that integration.

A proof of this claim would also provide a proof of the algebraic independence of pi and e over Q. This is an open problem.

Re: HSC 2016 4U Marathon - Advanced Level Yep, of course. You can also very quickly show that P vanishes to second order at x=0, cutting down your work even more. Just putting pen to paper immediately and solving it the way I did took way less time than it did to type it up though, so I...

Re: HSC 2016 4U Marathon - Advanced Level Comparing degrees, 2n=4+n, so all solutions must be quartic. Take P=ax^4+bx^3+cx^2+dx+e and sub into the polynomial equation. Considering the lowest order term on both sides we get d=e=0. So...

Re: HSC 2016 4U Marathon - Advanced Level With the probability question: I think you need to be far more rigorous about how these days temperatures are "randomly" selected in order to have a well-posed question. Typically this involves a distribution. Saying things like "the temperature is...

Re: HSC 2016 4U Marathon - Advanced Level Also note that it is not a significant leap of faith to differentiate polynomials with complex coefficients/domain. For such objects differentiation can be viewed as a purely formal linear map that sends the monomial z^n to nz^(n-1). One can study...

Re: HSC 2016 4U Marathon - Advanced Level Still not quite right...in your (**) line, substituting z=0 would give p(t)= p(0) + At^qM(0). Also, there is a deeper problem going wrong in your assumption that p(z)-p(t)=(z^n-t^n)M(z). Your reasoning (presumably) was that for nonzero t, we...