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Re: MX2 2015 Integration Marathon Bump it so I can have a crack :).

Re: MX2 2015 Integration Marathon This question is based on your ability to approximate sums by integrals, hence its location in this marathon thread. Prove that there exist positive constants C_1,C_2 such that C_1\leq\frac{n!}{n^{n+1/2}e^{-n}}\leq C_2 for all positive integers n. (In...

Re: HSC 2015 4U Marathon - Advanced Level What have you been saying in your last few posts? I have not seen you say the thing you said in the last post earlier? No-one is disputing that t=2t => t=0, but that doesn't make your proof attempt any more valid...

All good. The rescaling part is completely unnecessary btw, just makes things look a tad nicer.

Re: HSC 2015 4U Marathon - Advanced Level Of course it doesn't, but you did not use this precise statement in your proof.

Re: HSC 2015 4U Marathon - Advanced Level Anyway, here is my proof. There are most likely cleaner ways to do it, and maybe even obvious direct improvements to my method, I feel a little slow today.

Re: HSC 2015 4U Marathon - Advanced Level Why would the number of "terms" change anything? f(x)=g(x) is the same equation as f(x)/2+f(x)/2=g(x), although the latter LHS has two "terms". I write with quotation marks because "terms" is a completely ambiguous word, as the above example shows...

Re: HSC 2015 4U Marathon i) \binom{n}{n_1}\binom{n-n_1}{n_2}\ldots\binom{n_m}{n_m}=\frac{n!}{\prod n_i!} by expanding out the binomial coefficients in terms of factorials. ii) This quotient is precisely the RHS and hence integral.

Re: HSC 2015 4U Marathon Well if n is in it's own group, we obtain a partitioning of the set {1,...,n} by partitioning the remaining n-1 numbers into k-1 nonempty subsets, hence the first term. If n isn't, that means we need to partition the remaining n-1 numbers into k sets, and then choose...

Re: HSC 2015 4U Marathon - Advanced Level No, I understood it, am just not very good at / don't really like combinatorics so I thought I would let someone else do it :p. I am busy atm but can have a crack later tonight or tomorrow. My guess is that it is a clever application of the pigeonhole...

Re: HSC 2015 4U Marathon - Advanced Level You have some inequalities the wrong way around, idk if your method works exactly but some tweaking might fix it.

Re: HSC 2015 4U Marathon - Advanced Level Cauchy-Schwartz states for real variables we have: (ax+by+cz)^2 <= (a^2+b^2+c^2)(x^2+y^2+z^2) with equality iff (x,y,z)=(ta,tb,tc) for some real c. Lots of proofs of Cauchy-Schwartz are on this site, so I will assume it. (Can provide a short proof...

57% raw isn't awful. unless you are sure you could do better with different units, and your atar requirements are looking like a close call, stick with it.

Is this what you were looking for?

I think so. By symmetry considerations, it suffices to show that for z > 0, and (x,y,z) a point on the hyperbolic paraboloid H=\{(x,y,z):z=\frac{x^2}{a^2}-\frac{y^2}{b^2}\} that there exists exactly two (up to rescaling) nonzero triples (u,v,w) such that the line...

Re: HSC 2015 4U Marathon - Advanced Level Lol, woke up with an idea that I think does it. Last post we showed that the subset S of k satisfying P(k) = k^3 consists of integers m with 100 - m a divisor of 999900. We know that P(x) - x^3 vanishes at k in S. So by the factor theorem we have...

Re: HSC 2015 4U Marathon - Advanced Level Okay cool, seems hard! Will spend more time on it tomorrow morning, but here is some first progress: Since P has integer coefficients, we must have k - 100 | P(k) - P(100) = k^3 - 100. But k^3 - 100 = (k - 100)(k^2 -100k + 10000) + 999900. So...

Re: HSC 2015 4U Marathon - Advanced Level Can you please clarify the exact question statement? Are these polynomials definitely required to have integer coefficients? Is it definitely P(100)=100? Are we meant to find the maximum number of solutions to f(k) = k^3 possible, or the largest...

Re: HSC 2015 4U Marathon A pretty easy but nice application of these ideas is to prove the differentiation formulae for the trig functions from first principles. Note that the differentiation formulae trivially imply the limit in Sy's original question, since \lim_{x\rightarrow 0}\frac{\sin...

Re: HSC 2015 4U Marathon To justify a statement drsoccerball used earlier without proof, I am going to prove the following: \sin(x)< x < \tan(x) for x\in (0,\pi/2). We do it geometrically (it is tempting to fall into the trap of using calculus, but things like the formulae for...