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I don't use textbooks at all so I'm not the best person to ask, I can say not to bother with 4U fitzpatrick, its a horrible book, but I think 3U fitzpatrick is alright. I'm not sure about Terry Lee's because I don't have it but I've heard good things from other people about it. I'm not sure...

Re: HSC 2013 4U Marathon $The product of the two zeros of the quartic equation$ x^4-18x^3+kx^2+200x - 1984= 0 $is$ \ \ -32 $Find$ \ \ k

That is an issue, in an exam, if I had enough time left in a 3U exam (which usually if you are fast enough one can finish the 3U exam in under an hour) and you have a protractor and compass with you, since the diagram is really bad and that its a 'find' question, I would construct the diagram...

I don't think its that long, there are only some core things that needs to be done 1. Establish the goal of proving BD is a tangent 2. Finding BD in terms of known pronumerals 3. Making a numerical equation for BD 4. Proving that the equation holds 2, 3 is quite fast though. But I do agree...

Check the thread, that was a good challenge.

$if$ 64 \sin^2 70 \sin^2 10 \sin^2 80 = 4\sin^2 70 - 2\sin 70 Divide by sin(70), convert sin 80 = cos 10, then use double angle formula 64 \sin 70 (\frac{1}{4} \sin^2 20) = 4\sin 70 - 2 Divide by 2, convert 1 of the sin(20) into cos(70) 8\sin 70 \sin 20 \cos 70 = 2\sin 70 -1 Use double...

Looking at the diagram, draw a circle around points A, B, E. Our goal is to prove that BD is a tangent to the circle, so we can satisfy angles in alternate segment, finding that x=\angle BAE = 20^{\circ} To prove that BD is a tangent, we have to show that the segments, BD, DE, and DA...

Re: HSC 2013 4U Marathon $A recurrence relation is given by$ \ \ f_1=f_2=1 f_n = f_{n-1} + f_{n-2} $Find the sum$ \sum_{n=2}^{\infty} \frac{1}{f_{n-1} f_{n+1}}

Re: HSC 2013 4U Marathon I thought so too, its problem A1 for 2002 iirc.

Re: HSC 2013 4U Marathon Yep =)

Re: MX2 Integration Marathon Yeah I do quite like those substitutions, it seems so elegant. Like those random ones people think of like x=(1-u)/(1+u) or whatever \int_0^{\pi /2} \frac{\sin(2013x)}{\sin x} \ dx

Re: HSC 2013 4U Marathon Another good Putnam question $Let$ \ k \ $be a positive integer.$ $The$ \ n$-th derivative of$ \ \ \frac{1}{x^k-1} \ \ $has the form$ \frac{P_n(x)}{(x^k-1)^{n+1}} \ \ $, where$ \ \ P_n(x) \ \ $is a polynomial.$ $Find$ \ \ P_n(1)

Re: HSC 2013 4U Marathon The first answer is correct, as for the second one, let me check my proof again, I got the result from 'Art of Problem SOlving' and it asked whether it diverged and converged. Feel free to post your proof of the upper and lower bounds though EDIT: Yeah my proof is...

Re: HSC 2013 4U Marathon Yep that is correct. $What is the largest integer$ \ n \ $for which$ \ n^3+100 \ $is divisible by$ \ n+10 \ $?$

Re: HSC 2013 4U Marathon $Take the set of all positive integers$ \ S \ $which contains all positive integers without a zero in their representation$ $S = \{1,2, \dots , 9, 11, 12, \dots , 19, 21, \dots \} $Let each element in the set$ \ S \ $be$ \ \ a_k \ , \ k=1,2, \dots $Consider the...

Lets first deal with the first 100 digits. For 1,2,3,4,5,6,7,8,9 9 digits used for 9 pages. Next, 10 -> 99 90 pages, 90*2 = 180 digits used. So now we enter 3 digits, already having used 189 digits. So 875-189 = 686 digits left. That means 686 / 3 = 228.667 pages So if we go from 100...

Re: MX2 Integration Marathon \int \frac{x^2-1}{x(x^2+1)(\ln (x^2+1) - \ln x)} \ dx

Re: HSC 2013 3U Marathon Thread Without induction, we take the sum from 1 to n of: k(k+1)! - (k-1)k! = (k^2+1)k!

Are you sure there is no typo? x=pi/2 does not yield equality

Re: HSC 2013 3U Marathon Thread Yep well done $Show that$ \sum_{k=1}^n (k^2+1)k! = n(n+1)!