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Re: HSC 2015 4U Marathon - Advanced Level I think my method was an induction, but might have involved contradiction in proving one of the steps. Not sure about whether or not you can avoid induction.

As always, grinding at SSBM. Melbourne is hosting a major next month that has attracted several BIG names from overseas. It is unlikely there will be a single Australian in the top 8 lol. But I cannot wait to meet/play/talk with some of the best players on the planet of my favourite game :).

Re: MX2 2015 Integration Marathon Bump. Note that the upper bound has already been proved by Integrand above.

Re: MX2 2015 Integration Marathon Yep, so this establishes the upper bound. Geometrically this is the same as noting that we are really integrating around the unit circle, and our integrand is maximised at the point on the unit circle closest to z, then replacing the whole integrand by...

Re: MX2 2015 Integration Marathon I should clarify that you need to prove the existence of constants such that these inequalities hold for ALL z not on the unit circle. Your constants cannot depend on z themselves! (Otherwise these inequalities don't really tell you how fast the integral grows...

Re: MX2 2015 Integration Marathon Here is a question that concerns inequalities arising from integration. (It is harder conceptually than most questions in this thread, but easier than a lot of them in terms of how technically demanding the required manipulations are.) $Let $\alpha>0$ be a...

Re: MX2 2015 Integration Marathon I just did the shift by one (intuition told me this substitution would make it nicer because of the numerator/denominator symmetry) then multiplied the numerator and denominator by the numerator, to get rid of square roots up there. Alternatively you can get...

Re: MX2 2015 Integration Marathon I=\int_1^2 \sqrt{\frac{2-x}{2+x}}\, dx \\ = \int_1^2 \frac{2}{\sqrt{4-x^2}}-\frac{x}{\sqrt{4-x^2}} \, dx\\ \\ \\= \ldots (I am sure you can do it from here.)

Re: MX2 2015 Integration Marathon I_n=I_{n-1}+\int_0^{2\pi}(1+\cos(\theta))^{n-1}\frac{d}{d\theta}(\sin(\theta))\, d\,\theta\\ = I_{n-1}+(n-1)\int_0^{2\pi}(1+\cos(\theta))^{n-2}\sin^2(\theta)\, d\theta \\ = I_{n-1}+(n-1)\int_0^{2\pi}(1+\cos(\theta))^{n-1}(2-(1+\cos(\theta)))\, d\theta \\ =...

Re: HSC 2015 4U Marathon - Advanced Level $A subset $X$ of the plane is said to be \emph{convex} if the line segment joining any two points in $A$ lies entirely within $A$.\\ Given that $X_1,X_2,...,X_n$ are convex sets in the plane ($n\geq 3$) such that the intersection of any three of these...

Re: MX2 2015 Integration Marathon Here is a primitive for e^{x^2}, defined on the whole real line: \sum_{k=0}^\infty \frac{x^{2k+1}}{k!(2k+1)}.

Re: HSC 2015 4U Marathon - Advanced Level As a sidenote for any HS students that want to know where the extreme value theorem comes from, here is what you need to know for one proof. 1. The least upper bound property of the reals. (Any set A of real numbers which has an upper bound M has a...

Re: HSC 2015 4U Marathon - Advanced Level I think any proper proof of this will need a precise definition of the reals, which is comfortably outside the scope of mx2. (Unless you allow people to assume the extreme value theorem, which implies the claimed result almost trivially).

Typo.

Re: HSC 2015 4U Marathon A sequence converges to blah if the limiting value of the sequence is blah. The formal definition involves topological spaces.

Remember the rule z\overline{z}=|z|^2? This implies for nonzero z: z^{-1}=\frac{\overline{z}}{|z|^2}. In particular, inverting is the same thing as taking conjugates on the unit circle. For example \overline{w}=w^{-1}=w^5/w=w^4 if w is a 5-th root of unity.

Definitely. The question doesn't make sense as is, the conjugate root theorem tells us nothing if one of the roots is real. Beta can be absolutely anything.

Three column vectors are linearly dependent if and only if the matrix with these three columns has vanishing determinant.

Here is a cute little convergence lemma that gets used a lot in dynamics (eg in the study of entropy). A good exercise for students learning real analysis for the first time. Given a sequence (x_n) of positive real numbers such that x_{m+n} \leq x_n+x_m for every pair of positive integers m...

Re: HSC 2015 4U Marathon Newton's method (as taught in HS) definitely isn't good enough to prove inequalities like the one Sy123 posted without additional work. You are not taught any criteria by to check whether the resulting sequence does indeed converge to an exact root for given seed...