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Re: HSC 2013 4U Marathon That used to be my fastest method. I just thought of something arguably faster though. Let a=z1-z2, b=z2-z3, c=z3-z1. Observe that a+b+c=0 always. The equation given is equivalent to a^2+b^2+c^2=0. But since a+b+c is also zero, this is equivalent to the...

Re: HSC 2013 4U Marathon Also, more interestingly, the condition is equivalent to the three points forming an equilateral triangle. (If and only if).

Re: HSC 2013 4U Marathon remove modulus signs.

Re: HSC 2013 4U Marathon Well if the constant is small enough the locus will just be a point, or even nonexistent.

Re: HSC 2013 4U Marathon Correct answer, although I lost track of your logic halfway through. Will read more carefully later, it sounds like you knew what you were doing.

Those people don't know what they're talking about.

Re: HSC 2013 4U Marathon I think you probably use something equivalent to it whenever you prove something like this by induction, without really thinking about it in that way.

Re: HSC 2013 4U Marathon So it is obvious how you WANT students to do this question so I won't write that solution up. But just a neat observation, the notion of a sequences "derivative" from http://community.boredofstudies.org/showthread.php?t=292775 can be used to solve this question kind of...

Look up wikipedia for a rigorous definition of what 'lim' means and the question will become trivial. Roughly speaking, \lim_{x\rightarrow \infty}f(x)=L means that f(x) gets arbitrarily close to L when x is large. But here f(x) is in fact EQUAL to 0 everywhere (except at x=0), so certainly...

Re: HSC 2013 4U Marathon Square root magically disappeared.

Re: HSC 2013 4U Marathon By setting a=b=1 your answer has the revolutionary consequence that the unit circle has area 8/3.

No worries, I feel like I did something slicker last time but whatever...its late and this works.

$We have that the derivative of a polynomial which has a root of multiplicity $n>1$ at $x=\alpha$ has a root of multiplicity $n-1$ at $\alpha$. Applying this at $0$ and $1$ (observing that $\frac{d}{dx}(\phi(x)-1)=\phi'(x)$) gives us:\\ $\phi'(x)=Ax^2(x-1)^2.$ for some unknown nonzero constant...

Low-cus. I don't get laughed at about it though.

Re: HSC 2013 4U Marathon Good. You will have no trouble with c). The part of this problem I consider 'interesting' is that people's intuitions may fool them for a and b. In a vaguely similar style I made up the following question based on something a friend asked me during the US election...

Re: HSC 2013 4U Marathon Fun little question. a) A whole grade consisting of several (>1) classes participates in a mathematics competition. In each class the average female score is higher than the average male score. Is it possible for the average male score over the entire grade to be...

What is the derivative of g(x) + a - x?

I know right, mostly usyd people so its a little less surprising but still.

https://www.facebook.com/skgomes add me yo.

Drinking with bos people sounds tres fun, are onesies permitted/encouraged?