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Re: HSC 2015 4U Marathon Rather use x \geq \ln (1+x) Intuition behind this transformation is that we want an inequality with an ln() function in it because the RHS has a ln() function

Re: HSC 2015 4U Marathon \\ $Let$ \ a,b \ $be positive real numbers so that$ \ a + b = 1 \\ \\ $Prove that$ \ \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2}{3}

Re: HSC 2015 4U Marathon Transform the inequality in part (i), as it is in the form now, it is useless for part (ii)

There are some techniques that would let you 'see' the proof to a problem more easily For instance, you should be familiar with every form of the AM-GM inequality, i.e. a^2 + b^2 \geq 2ab (a+b)^2 \geq 4ab a + \frac{1}{a} \geq 2 a^3 + b^3 \geq ab(a+b) a^2 + b^2 + c^2 \geq ab...

Re: HSC 2015 4U Marathon \\ $i) Prove that$ \ e^x \geq x + 1 \ $for$ \ x \geq 0 \\ $ii) Use the above inequality to prove that$ \ \sum_{k=1}^n \frac{1}{k} > \ln (1 + n) \\ $iii) Hence prove that (where$ \ a \neq 0) \ \sum_{n=1}^{N} \frac{1}{an + b} \to \infty \ $as$ \ N \ $increases without...

Re: HSC 2015 4U Marathon An easier question: \\ $The polynomial$ \ z^3 - Az^2 + Az - 1 = 0 \ $has three roots of equal modulus, find the range of real values$ \ A \ $can take.$

Re: HSC 2015 4U Marathon I think at least part (i) could be, and maybe part (ii) if split into different parts

Re: HSC 2015 4U Marathon Here is a my HSC style solution: \\ $Watch the clock from time$ \ k : 00 \ $to$ \ k:np-1 \ $with$ \ k = 0, 1, \dots, p - 1 \\ $At the start of the watch, the hour-hand points to the$ \ nk^{th} \ $mark on the clock, and the minute-hand points to the$ \ 0^{th} \ $mark...

Re: HSC 2015 4U Marathon InteGrand has pointed out an error in the question, for part ii.a there are (p-1) straight times, not p straight times, edited ------------------------- \\ $A clock-face has a hours hand and a minutes hand, there are marks around the clock for each hour and each...

Re: HSC 2015 4U Marathon I got a straight time for every hour, so p straight times PM me your working out

Re: HSC 2015 4U Marathon n*p minutes ^^

Re: HSC 2015 4U Marathon Here is a question I thought up while staring at a clock waiting for the time to pass: \\ $A clock-face has a hours hand and a minutes hand, there are marks around the clock for each hour and each minute. A specific time is 'straight' if the minute-hand and the...

Re: HSC 2015 4U Marathon \\ $Consider the polynomial$ \ p(x) = x^3 - 3x + 1 \\ \\ $i) Transform the equation into a function of$ \ t \ $using the substitution$ \ x = t + \frac{1}{t} \\ \\ $ii) Hence or otherwise, show that$ \ x = -2\cos \frac{\pi}{9} \ $is a root of the polynomial$ \ p(x)

it's a Friday, I went last year but it started late and I had to leave halfway through

ヽ༼ຈل͜ຈ༽ﾉ UPLOAD THE PAPER OR RIOT ヽ༼ຈل͜ຈ༽ﾉc

Something like this: ?

What does a feminist wanting the right to custody, abortion and birth control have to do with the crippling egocentrism of the modern psyche? He is simply injecting his own personal invective against feminism to taint a video that otherwise contains some nuggets of wisdom (viz. egocentrism). We...

Re: HSC 2015 4U Marathon correct, but can you give the reason why? (unless it was already posted)

Re: HSC 2015 4U Marathon \\ $Let the number of ways to move up a staircase of$ \ n \ $steps, where one can only move up 1 or 2 steps at a time, be$ \ s_n \\ $Find a recurrence formula for$ \ s_n \ $and prove its correctness$

The statement "Jesus and God differ" is quite clear, it would mean that Jesus has a property that God does not. It could be a property of the essence that is the problem (as I outlined before, Jesus isn't triune, yet God is triune, a difference in essence), or it could be a property of action...