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Trinitarians hold the following set of propositions: 1. The Father is God 2. Jesus is God 3. The Father is not Jesus Where 'is' is used in the sense of identity, i.e. The Father is identical with God But of course, the 3 propositions that I've layed out are self-evidently inconsistent.

Post good poems here for reflection! All human beings are members of one frame, Since all, at first, from the same essence came. When time afflicts a limb with pain The other limbs at rest cannot remain. If thou feel not for other’s misery A human being is no name for thee. - Saadia Shirazi

Re: HSC 2015 4U Marathon - Advanced Level (taking the word degree to describe the number of lines that goes through a specific intersection point) It is sufficient to prove that there is at least one intersection point of degree 2, because if there is, one can just remove a line that goes...

Re: HSC 2015 4U Marathon - Advanced Level I hope this is valid \\ $Suppose that the plane$ \ P_n \ $has some arrangement of lines no pair of which is parallel, and they do not all pass through a single point, and let the number of intersection points be at most$ \ n-1 \\\\ $Let the...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Suppose there is a polynomial$ \ P(x) = 1 + x + x^2 + \sum_{k=3}^n a_kx^k \ $and all of the roots$ \ r_1,r_2,\dots,r_n \ $are real$ \\ \\ $Then,$ \\\\ \prod_{k=1}^n r_k = \frac{1}{a_n} \ (*) \\ \\ \sum_{k=1}^n \frac{\prod_{j=1}^n r_j}{r_k} =...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Prove that there is no integer$ \ n \ $so that:$ \ (n-3)^3 + n^3 = (n+3)^3

Re: HSC 2015 4U Marathon \\ $By using de Moivre's theorem and the Binomial Theorem, show that$ \\\\ \sum_{i=0}^n \binom{n}{i} \sin((2i+1)x) = 2^n \cos^n x \sin((n+1)x)

Re: HSC 2015 4U Marathon I may be wrong but I feel like there is some sort of circularity here: It might be a misinterpretation but it seems you assume that x(0) is the lowest of the x(n) which from there allows you to prove that x(n+1) > x(n). So it is either circular in that you...

Re: HSC 2015 4U Marathon \\ $Let$ \ x_0 = 4 \ $and$ \ x_{n+1} = \frac{1}{5}(x_n^2 + 6) \\ $i) Show that$ \ x_{n+1} > x_n \ $for integers$ \ n \geq 0 \\ $ii) Show that if$ \ x_0 < 3 \ $then it is not true that$ \ x_{n+1} > x_n \ $for integers$ \ n \geq 0

Re: HSC 2015 4U Marathon 675 = 3^3 5^2 = A^2 - B^2 \\ $The solutions to this equation are of the form$ \\ A - B = 3^{n}5^{m} \ $and$ \ A+B = 3^{3-n}5^{2-m} \ $for$ \ n = 0, 1, 2, 3 \ $and$ \ m = 0, 1, 2 \\ $Since$ \ A = \frac{3^{n} 5^m + 3^{3-n} 5^m}{2} \ $and$ \ B = \frac{3^{3-n} 5^{2-m}...

Re: HSC 2015 4U Marathon Can you expand on your method? From what I can see you've proven that there are 2 possibilities, for a, b (that a is even b odd, a odd b even) but that doesn't mean there are exactly 2 solutions, for example 675 can be written in (i think it's 6) different ways of...

Re: HSC 2015 3U Marathon \\ $i) Show that$ \ \frac{(1+x)^{n+1} -1}{n+1} = \sum_{k=0}^n \frac{1}{k+1} \binom{n}{k}x^{k+1} \\ $ii) Hence show that$ \ \sum_{k=0}^n \frac{1}{(k+1)(n-k+1)} \binom{n}{k} \binom{n}{n-k} = \frac{1}{(n+1)^2} \binom{2n+2}{n+2}

Re: MX2 2015 Integration Marathon I think that's with seanieg's question

Re: MX2 2015 Integration Marathon \\ \int_{1/t}^t \frac{\tan^{-1}x}{x} \ dx \ $for$ \ t > 0

Re: MX2 2015 Integration Marathon I wasn't able to think of a way to approximate the lower bound as \sqrt{2\pi} (though technically if you allow for a strict lower bound like 2pi as an answer to the question, then a strict lower bound of 1 would be sufficient too and easy to prove, though...

Re: HSC 2015 4U Marathon I'm getting a lot of these questions from: http://mathsorchard.weebly.com/step-past-papers.html http://www.physicsandmathstutor.com/step/ STEP papers are very valuable resources for MX2 since the questions are of advanced difficulty (relative to the HSC) and are of...

Re: HSC 2015 4U Marathon \\ $For (ii), suppose that there are in fact$ \ F_i \ $and$ \ F_j \ $with$ \ i < j \ $such that they have a common factor$ \ M > 1 \\ \\ $Then, by the identity in part (i)$ \ F_0F_1\dots F_i \dots F_{j-1} = F_j - 2 \\ \\ $By dividing both sides by$ \ M \ $then$ \ F_0...

Re: HSC 2015 4U Marathon \\ $Find the number of ways of writing the number 675 as the difference of 2 squares$

Re: HSC 2015 4U Marathon \\ $Show that if$ \ p, q \ $are both distinct prime numbers greater than 2, then$ \ pq \ $can be written as the difference of two positive squares, in exactly two different ways$ \\ \\ $Show that this cannot be the case if either$ \ p \ $or$ \ q \ $equals 2$.

Re: HSC 2015 4U Marathon It's a little unclear what you are saying in the middle but I see what you are trying to say, it is my method too