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Re: HSC 2014 4U Marathon Wow haha, my way is much much longer but uses more basic inequalities

Re: HSC 2014 4U Marathon This is a good inequality: \\ $Consider the positive real numbers$ \ a_1 , a_2 , \dots , a_n \ $such that$ \\ a_1 + a_2 + \dots + a_n = 1 \\ $Show that$ \\ \\ \sum_{k=1}^n\frac{a_k}{\sqrt{1-a_k}} \geq \frac{1}{\sqrt{n-1}} \sum_{k=1}^n \sqrt{a_k}

Re: HSC 2014 4U Marathon This is a much more better version of my proof lol (I too went for a 3 inequality)

Re: HSC 2014 4U Marathon The smallest gap yes, I mean that the smallest gap between any 2 rational numbers is 1/(common denominator). Take arbitrary rationals with integer variables \frac{a}{b} - \frac{c}{d} \frac{ad}{bd} - \frac{bc}{bd} = \frac{ad-bc}{bd} In order to minimize the gap...

Re: HSC 2014 4U Marathon Hopefully this is valid, $Consider 2 rational numbers with the same denominator, in the form$ \ \ \frac{p}{r} \ $and$ \ \frac{q}{r} \ \ $which can be done with any 2 rational numbers$ $Let$ \ q > p \ $be integers$ $And so the lowest gap between the two distinct...

Re: HSC 2014 4U Marathon I guess I have some reading to do haha

Re: HSC 2014 4U Marathon Ah ok I see the rational one, what do you mean proving M_p to be continuous? How would we do that?

Re: HSC 2014 4U Marathon For the first problem, I will attempt to prove it for integers p, q, hopefully then someone else can extend it to the reals somehow until I try to figure it out \\ $We must prove that$ \ M_p \leq M_q \ $for now we assume that$ \ p,q \in \mathbb{N} \ $We will attempt...

Well done! Make sure to try state rank for Mx1/Mx2 next year =)

I highly recommend Matthew for tutoring, he is highly sociable yet knows what he is talking about and very knowledgeable in Mathematics. It is quite clear from just a bit of browsing of the Mathematics forums that Matthew indeed possesses a desire to help people and explain mathematical concepts...

Re: HSC 2014 4U Marathon That is the general idea for all 3 haha (and a bit more)

Re: HSC 2014 4U Marathon \\ $Suppose for positive real$ \ a,b,c,d \ \ $the polynomial$ \\ p(x) = x^4 - 4ax^3 + 6b^2x^2 - 4c^3x + d^4 \ \ \ \ $has 4 distinct positive real roots$ \\ $Show that$ \\ \\ $i) $ \ c>d \\ \\ $ii)$ \ b >c \\ \\ $iii)$ \ a>b

Re: HSC 2014 4U Marathon Alternatively: $Consider the function$ \ f(x) = 8(x^4+1) - (x+1)^4 $Show that$ \ f(x) \geq f(1) = 0 $Hence$ \ \ 8(x^4+1) \geq (x+1)^4 $Let$ \ x = \frac{a}{b}

Re: HSC 2014 4U Marathon $Prove for real$ \ a,b \ \ 8(a^4+ b^4) \geq (a+b)^4

Try and send now :)

thanks guys!

:D

Nah it just puts my chances at state rank much lower :/

:(

State Rank is all 1st to 10th (or 20th or 5th). First in State is just first.