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Re: HSC 2013 4U Marathon \left(\frac{1}{x}+\frac{1}{y}\right)^3=\frac{1}{x^3}+\frac{1}{y^3}+3\left(\frac{1}{x^y}+\frac{1}{xy^2}\right) -1=\frac{x^3+y^3}{x^3y^3}+\frac{3}{xy}\left(\frac{1}{x}+\frac{1}{y}\right) -1=\frac{4}{x^3y^3}-\frac{3}{xy} -x^3y^3=4-3x^2y^2 Let z=xy...

You have ax+bx=40 and a+x=2(b+x) Using the second equation express x in terms of a and b. Factorise the first equation and sub in for x. Now think about possible factors of 40. Second question: ab+ba (where a and b are digits of your two digit number). Your sum is then 10a+b +...

Re: HSC 2013 3U Marathon Thread Is everyone allowed to answer these questions?

Thanks again. :)

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:) In that case let angle BAX=\alpha and we easily get the following angles. XAC=60-\alpha ACX=60+\alpha Without loss of generality let AB=AC=BC=1. Then, using the sine rule BX=\frac{\sin \alpha}{\sin 60} XC=\frac{\sin(60- \alpha)}{\sin 60} AX=\frac{\sin(60+...

Are you allowed to refer to Ptolemy's theorem without proof? If so question 1b becomes trivial.

If you just focus on \displaystyle \frac{d}{dx}(3x-2)^5 for a moment.. This is a (simple) example of the chain rule. The 3 is the derivative of 3x-2.