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Re: HSC 2015 4U Marathon \\ $Show that 1 less than the square of an odd number is divisible by 8$

Re: HSC 2015 4U Marathon First of all, 3^{n_1} 5^{m_1} = 3^{n_2} 5^{m_2} \Rightarrow n_1 = n_2 , m_1 = m_2 Now, if we want to find a number that divides into 3^n5^m , we need to be able to construct this number by multiplying 3's and 5's. What I mean is, \\ $if$ \ k \ $divides$ \ 3^n5^m \...

Re: HSC 2015 4U Marathon \\ $Find the amount of positive integers that divide into$ \ 3^n 5^m \ $for non-negative integers$ \ m,n

Re: HSC 2015 4U Marathon - Advanced Level \\ $Find the set of all positive integers$ \ n \ $for which$ \ n \ $does not divide$ \ (n-1)!

Re: HSC 2015 4U Marathon - Advanced Level What are you doing? This is a marathon, not a court, post a question.

Re: HSC 2015 3U Marathon

Re: MX2 2015 Integration Marathon \\ \int x^2(\sin \ln x + \cos \ln x) \ dx

Re: HSC 2015 4U Marathon \\ $Let$ \ S(n,k) \ $be the number of ways to divide the collection of numbers$ \ \{1,2,3,\dots,n \} \ $into$ \ k \ $non-empty groups$ \\ $e.g.$ \ S(3,2) = 3 \ $since one can divide$ \ (A,B,C) \ $into groups$ \ (AB, C) , (AC, B) \ $and$ \ (BC, A) \\ $By considering...

Re: HSC 2015 3U Marathon Neither of those are correct: http://www.wolframalpha.com/input/?i=%28x%2B1%2Fx%29%5E3+%281%2Bx%29%5E5 \left(x + \frac{1}{x} \right)^3 (1+x)^5 = \frac{1}{x^3} (1+x^2)^3(1+x)^5 \\ $So, instead of finding the coefficient of$ \ x^2 \ $we can find the coefficient of$...

Re: HSC 2015 4U Marathon \\ $Let$ \ n_1,n_2, \dots , n_m \ $be non-negative integers such that$ \ n_1 + n_2 + \dots + n_m = n \\ \\ $i) How many ways can one distribute$ \ n \ $chickens into$ \ m \ $pens, with pen$ \ i \ $having$ \ n_i \ $chickens for$ \ i = 1,2,\dots,m \\ \\ $ii) Hence show...

Re: HSC 2015 4U Marathon I think there are some typos there but I see what you're trying to say, what I did: x^2 + y^2 \geq 2xy \Rightarrow (x+y)^2 \geq 4xy \Rightarrow \ \frac{x+y}{2} \geq \frac{2xy}{x+y} \\ \therefore \ \frac{2ab}{a+b} + \frac{2ac}{a+c} + \frac{2bc}{b+c} \leq...

Re: HSC 2015 3U Marathon

Re: HSC 2015 3U Marathon That works (if you did your differentiation correctly), but I would first consider the function f(x) = (1 + 1/(x+1))^{x+1} where x is among the reals, and differentiate that since you can't differentiate over the integers! The method I was looking for is: \left(1 +...

Re: HSC 2015 4U Marathon - Advanced Level nice method, mine was a little longer: x = \frac{a}{a+b+c}, \ y = \frac{b}{a+b+c}, \ z = \frac{c}{a+b+c} \\ \sum_{cyc} \frac{\frac{a}{a+b+c}}{1 - \frac{a}{a+b+c}} = \sum_{cyc} \frac{x}{1-x} \\ f(x) = \frac{x}{1-x} \geq \frac{9}{4}x - \frac{1}{4}...

Re: HSC 2015 3U Marathon \\ $i) Expand using the Binomial Theorem$ \ \left(1 + \frac{1}{k+1} \right)^{k+1} \ $for positive integer$ \ k \ $and prove that$ \ \left(1 + \frac{1}{k+1} \right)^{k+1} > 2 \\ \\ $ii) Hence or otherwise, prove using mathematical induction$ \ \left(\frac{n+1}{2}...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Prove$ \ \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \geq \frac{3}{2} \ $for positive$ \ a,b,c

Re: HSC 2015 4U Marathon \\ $i) Show that$ \ x^2 + y^2 \geq 2xy \ $for positive real$ \ x,y \\ $ii) Hence show that$ \ \frac{2ab}{a+b} + \frac{2ac}{a+c} + \frac{2bc}{b+c} \leq a + b + c \ $for positive real$ \ a,b,c (note, harder than the previous couple, but still HSC exam level)

Re: HSC 2015 4U Marathon - Advanced Level Which chord is this? I just let the fractional part be 'r' and found a quadratic equation that implied r < 1/4 (after doing what you did first)

Re: HSC 2015 4U Marathon \\ $Let$ \ S_n(\alpha) = \sum_{k=1}^n \frac{1}{k^{\alpha}} \ $where$ \ \alpha > 0 \\ $i) Show through a graph or otherwise$ \ \int_1^{n-1} \frac{1}{x^{\alpha}} \ dx < S_n(\alpha) < 1 + \int_1^n \frac{1}{x^\alpha} \ dx \\ $ii) Hence show that$ \ \lim_{n \to \infty}...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Prove that the fractional part of$ \ \sqrt{4n^2 + n} \ $is less than$ \ \frac{1}{4}