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Re: HSC 2016 4U Marathon Three mathematics students were seated around a table each wearing a coloured hat, either red or yellow, given to them by their teacher. The students could not see the colour of their own hat, but could see the colour of the other students. The teacher told the students...

Re: HSC 2016 4U Marathon \\ $Find with proof$ \ \lim_{x \to 0} \frac{\sin(x^2)}{(\sin x)^2}

Re: HSC 2016 4U Marathon \\ $i) Find a closed formula for$ \ \int_0^{\pi /2} \sin^{2n+1}(x) \ $d$x \ $where$ \ n \ $is a non-negative integer$ \\\\ $ii) Hence find$ \ \sum_{n=0}^{\infty} \frac{1}{\binom{2n}{n}} EDIT: Will rework so it fits into the non-advanced marathon

Re: HSC 2016 4U Marathon - Advanced Level \\ $The proposition in question can be split into two individual propositions$ \\ \textbf{(1)} $ Prove that polynomials of the form$ \ Q(S_1,S_2,\dots, S_n) \ $are symmetric$ \\ \textbf{(2)} $ Prove that for polynomials$ \ P \ $that are symmetric...

Re: MX2 2016 Integration Marathon \\ $Find$ \ \int_{-\pi}^{\pi} \sin(nx) \sin(mx) \ dx \ $in the cases where$ \ m \neq n \ $and$ \ m = n \ $and$ \ m,n \ $are integers$

Re: MX2 2016 Integration Marathon \\ $i) Prove that$ \ \int_0^a f(x) \ dx = \int_0^a f(a-x) \ dx \\ $ii) Hence find$ \ \int_0^{\pi} \frac{x \sin x}{1 + \sin^2x } \ dx

Re: HSC 2016 4U Marathon Bringing it back to more familiar ground: \\ $By plotting a rhombus onto the complex plane and using complex numbers, prove that their diagonals are perpendicular to each other$

Re: HSC 2016 4U Marathon \\ $Given$ \ f''(x) > 0 \ $for all$ \ a \leq x \leq b \\ $Then$ \ f'(x) \ $is a monotone increasing function on$ \ a \leq x \leq b \\ $Then, for$ \ a \leq x, p \leq b \ $we have$ \ x \geq p \Rightarrow f'(x) \geq f'(p) $, and we have$ \ x \leq p \Rightarrow f'(x)...

Re: HSC 2016 MX2 Combinatorics Marathon s(m,k) = c_ms(m-1,k) + (1-c_m) s(m-1,k-1)

For those who are curious, the arithmetical manipulations preserve the truth of the inequality if and only if the function being applied is monotone increasing across the domain of the inequality. If it was monotone decreasing on the other hand, the inequality sign 'flips', in fact this is why...

Re: HSC 2016 4U Marathon For questions of an appropriate difficulty and beyond the capabilities of most current 2016er 4U students (at this stage in which they only know Complex Numbers/Polynomials), take it to the Advanced Thread. ----------------------- This is at a reasonable difficulty...

I'd notably mention that in UNSW itself there is the "Sydney School" of the philosophy of mathematics, namely an Aristotelian realist approach to Mathematics http://web.maths.unsw.edu.au/~jim/structmath.html

Need measurements that are infinitely precise be possible if mathematical entities exist? It seems to me that they are different questions. Right I see that argument, I had arguments of the form that seek to prove that if p/q = sqrt(2) then p and q are both even which yields contradiction, in...

The philosophy of mathematics asks questions about the foundations of mathematics, validity of certain truths in mathematics, it's assumptions, implications, the correspondence of mathematics and reality and so on. One primary question to ask is whether mathematics is a subject that refers...

Re: MX2 2016 Integration Marathon Yep the second term on the numerator is just 3/2, changed, (didn't affect rest of the working)

Re: MX2 2016 Integration Marathon \\ = \int \frac{1+\sin x + \sin^2 x + \sin^3x}{(1+\cos^2x)(1+\cos x)} \\ = \int \frac{2 - \cos^2x}{(1+\cos x)(1+\cos^2x)} \ dx + \int \sin x \frac{2 - \cos^2x}{(1+\cos x)(1+\cos^2x)} \ dx \\ $Using partial fractions$ \ \frac{2-A^2}{(1+A)(1+A^2)} =...

Re: HSC 2016 4U Marathon Well done -------- A more intuitive way of approaching part (ii) especially is to ask, "what does it mean for every complex number to be represented?" Essentially, it just means that every complex number's argument and modulus can be represented, and the domains for...

Re: HSC 2016 4U Marathon - Advanced Level Yep well done My approach was \\ p_{n} = \frac{1}{2n+1} (1-p_{n-1}) + \frac{2n}{2n+1}p_{n-1} \ $(same reasoning as you use)$ \\ (2n+1)p_n - (2n-1)p_n = 1 Then the sum telescopes and we get n/(2n+1) It's pretty much induction in disguise

Re: HSC 2016 4U Marathon - Advanced Level \\ $Let there be coins$ \ C_k \ $for each positive integer$ \ k \ $where the probability$ \ C_k \ $returns heads is$ \ \frac{1}{2k+1} \\ $Let the probability that flipping the first$ \ n \ $coins returns an odd number of heads in total be$ \ p_n \\...

Re: HSC 2016 4U Marathon - Advanced Level \\ L = \int_0^1 (1-\sqrt[p]{x})^q \ dx \\ x = u^p \ \Rightarrow \ L = \int_0^1 pu^{p-1} (1-u)^q \ du \\ $By integrating by parts$ \ L = u^p(1-u)^q|_0^1 + \int_0^1 q(1-u)^{q-1} u^p \ du \\ \Rightarrow \ L = \int_0^1 q(1-u)^{q-1} u^p \ du \\ v =...