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Re: HSC 2014 4U Marathon - Advanced Level \\ $Find$ \ \ \sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}

But that was a genius move by Holland, the GK probably studied Costa Rica's penalties as well

I only watched the Germany vs France and Argentina vs Belgium games. Dissapointed in Belgium tbh, and I feel the same way about Germany and France Though its expected for teams to play cautious at this level unlike the group stage since there is much more at stake I think.

My interpretation of the question: What powers of 2 are divisors of 10! ? 10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 = (2\cdot 5) \cdot (3^2) \cdot ( 2^3) \cdot 7 \cdot (2 \cdot 3) \cdot5 \cdot (2^2) \cdot 3 \cdot 2 \\ \\ 10! = 2^7 \cdot 3^4 \cdot 5^2 \cdot 7...

Re: HSC 2014 4U Marathon - Advanced Level \\ $If$ \ a,b,c \ $are integers such that$ \ \ ab + bc +ac = 1 \ \ $show that$ \\ (1+a^2)(1+b^2)(1+c^2) \ $is a perfect square$

Re: MX2 Integration Marathon \int (x-1) \sqrt[3]{x-2} \ dx

Re: HSC 2014 4U Marathon - Advanced Level

Re: HSC 2014 4U Marathon - Advanced Level \\ $Let$ \ x = a+b-c \ , \ y = b + c - a \ , \ z = a+c - b \\ \\ \Rightarrow \ $Prove$ \ \frac{x+y}{2} \cdot \frac{x+z}{2} \cdot \frac{y+z}{2} \geq xyz \\ \\ $If$ \ x,y, z \ $are positive, (i.e. sides of a triangle)$ \\ \frac{x+y}{2} \geq \sqrt{xy} \ \...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Find a polynomial with integer co-efficients, with one of its roots being$ \ \left( \frac{5}{3} \right)^{1/6} + \left(\frac{3}{5} \right)^{1/6}

Re: HSC 2014 4U Marathon - Advanced Level Elegant solution, well done! Here is mine: \frac{a^2}{a+b} = \frac{a^2+ab-ab}{a+b} = a - \frac{ab}{a+b} \\ (a+b)^2 \geq 4ab \Rightarrow \ \frac{a+b}{4} \geq \frac{ab}{a+b} \\ \\ \Rightarrow \ a - \frac{ab}{a+b} \geq a - \frac{a+b}{4} \therefore \...

Thanks for the help, the answer is correct! Would rep, but need to spread (837799)

The 'count' for the number of iterations needed from 1 to 100 000 is only 350, for 1 000 000 I don't think it would go over 2^31 Likewise for the number being iterated Do you have suggestions on how to put more resources into running the program?

Hello all, I am posting this here because it has something to do with mathematics, but mostly programming, however the programming forums are mostly dead. This is regarding Problem 14 of Project Euler I decided to try and compute everything to see if I can brute force out the answer I...

Re: HSC 2014 4U Marathon Here is a proof I devised for the AM-GM inequality of the third degree \\ $i) Show that$ \ a^3 + b^3 \geq ab(a+b) \ $for positive$ \ a,b \\ \\ $ii) Show that$ \ a^3 + b^3 + c^3 \geq \frac{1}{2} (a(b^2+c^2) + b(a^2+c^2) + c(a^2+b^2)) \\ \\ $iii) Hence deduce that$ \\ \\...

\\ $For both your answers, if you multiply by$ \ -1 \ $you get the right answer, this is because$ \ a = - kv^3 \ $in the first, and in the second$ \ a = -k(v^2+c^2) \\ \\ $Why is that?$ \\ \\ $Since the force in question is a resistance, and thus is resisting movement, thus it has a hindering...

\\ $Parts (ii) and (iii) should come immediately from the recursion relation in (i)$ \\ \\ $Proceeding$ \\ \\ $(i)$ \ I_{n+1} = \int_1^2 \left(1 - \frac{1}{x} \right)^{n+1} \ dx \\ \\ u = \left(1 - \frac{1}{x} \right)^{n+1} \ , \ du = \frac{(n+1)}{x^2} \left(1 - \frac{1}{x} \right)^{n} \\ dv =...

Re: HSC 2014 4U Marathon Yep well done, this was my proof: \angle BPQ = \angle BCA \angle CBA = \angle PCQ (By angle in same segment) (Using A_{ABC} as the area of triangle ABC) A_{PBC} = A_{PBQ} + A_{PQC} \\ \frac{1}{2} PB \cdot PC \sin \frac{2\pi}{3} = \frac{1}{2} PB \cdot PQ \sin...

Australia getting their ass handed to them Is this the quality of football in the A-League as well?

Re: HSC 2014 4U Marathon You have made a mistake in your first line, Q is the intersection between the 2 chords, not P. So from your proof it now is: AQ x QP = BQ x QC -- I can reveal my proof if you wish, or I can let you have another go

Oh wow. I just have no words