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Re: HSC 2016 4U Marathon - Advanced Level \\ $An$ \ n \ $sided regular polygon is such that no side is vertical in gradient$ \\ $Prove that if$ \ m_k \ $is the gradient of the$ \ k^{\text{th}} \ $side, then:$ \\\\ m_1 m_2 + m_2 m_3 + \dots + m_{n-1}m_n + m_n m_1 = - n

Re: HSC 2016 4U Marathon - Advanced Level Your argument needs to be clearer for it to be a mathematical proof I'm still not sure what you are trying to say here, the equations (x-1)(x- (p+\sqrt(q)) = 0 and x - p - \sqrt{q} = 0 both have a root in common, but why does tha tell us anything...

Re: HSC 2016 4U Marathon - Advanced Level \\ $On a perfect circular billiard table, the ball is hit at angle$ \ \theta \ $radians away from the line joining the starting point and the centre$ \\ $Find the average distance of the ball to the centre in terms of$ \ \theta

Re: HSC 2016 4U Marathon - Advanced Level What do you mean by "both equations are equal"? How does this allow us to prove that P(p + sqrt(q)) = 0 -> P(p - sqrt(q)) = 0? (for any integer polynomials P)

Re: HSC 2016 4U Marathon - Advanced Level \\ $An integer polynomial is a polynomial where all the co-efficients are integers, let$ \ p(x) \ $be such an integer polynomial$ \\ \\ $i) Show that if$ \ p + \sqrt{q} \ $is a root of$ \ p(x) \ $then$ \ p - \sqrt{q} \ $is a root, where$ \ p \ $is...

Yep my bad, didn't give too much thought to it

\\ $Suppose that there exists two points on the circle that are rational, let these points be$ \ P_1(p_1, q_1) \ $and$ \ P_2(p_2,q_2) \ $where$ \ p_1,p_2,q_1,q_2 \ $are rational$ \\ $The points lie on the circle$ \ (x-\pi )^2 + (y-e)^2 = r^2 \\ \Rightarrow \ [1]: \ (p_1 - \pi)^2 + (q_1 -...

Re: HSC 2016 4U Marathon - Advanced Level This may or may not be completely wrong and/or based on a misinterpretation of the question ------ Pr(John picks the right day) = Pr(John picks the first day, and it is the hottest the second day) + Pr(John picks the second day, and it is the...

Re: HSC 2016 4U Marathon - Advanced Level edit; nvm

Re: HSC 2016 4U Marathon - Advanced Level \\ $Show that there is no polynomial$ \ p(x) \ $that has integer coefficients and that there are three distinct integers$ \ a,b,c \ $such that$ \\\\ p(a) = b, \ p(b) = c, \ p(c) =a

Re: HSC 2016 4U Marathon - Advanced Level \\ $Find the minimum value of$ \ f(x) \ $for$ \ 0 <x < 1 \ $where$ \ f(x) = \int_0^{\frac{\pi}{4}} |\tan t - x| \ $d$t

Re: HSC 2016 4U Marathon - Advanced Level You're thinking of the factorisation of (t+h)^q + t^q \\ 1 + x + \dots + x^{q-1} = \frac{x^q -1}{x-1} \\ $Let$ \ x = \frac{t+h}{t} \ $and multiply both sides by$ \ t^{q-1}

Re: HSC 2016 4U Marathon - Advanced Level Revamping solution (and making some of the inferences more rigorous), had fun writing this \\ p(z(\cos \alpha + i \sin \alpha)) = p(z) \\\\ \Rightarrow \ p(z(\cos k\alpha + i \sin k\alpha)) = p(z) \ $for all integers$ \ k \geq 0 \\ \alpha \ $is...

Re: HSC 2016 4U Marathon - Advanced Level \\ p(z(cos \alpha + i \sin \alpha)) = p(z) \Rightarrow \ p(z(cos k\alpha + i \sin k\alpha)) = p(z) \ $for all integers$ \ k \geq 0 \\ \alpha \ $is either a rational multiple of$ \ 2\pi \ $or not, if it is not, then$ \ \cis k \alpha \ $is distinct for...

Re: HSC 2016 4U Marathon - Advanced Level Here is a good question (not too mechanical) \\ $Two polynomials$ \ P(x) \ $and$ \ Q(x) \ $have real co-efficients and are such that$ \ P(x) = Q(x) \ $for real values of$ \ x \\ $Prove$ \ P(x) = Q(x) \ $for all complex values of$ \ x

Re: HSC 2016 4U Marathon - Advanced Level For people unfamiliar with modular arithmetic: \\ N = 10^n a_n + \dots + 10a_1 + a_0 \ $is a representation of any number with digits$ \ a_n, a_{n-1}, \dots , a_0 \\ $Let$\ k = a_n + a_{n-1} + \dots + a_0 \\ N-k = \sum_{i=1}^n (10^i - 1)a_i =...

Re: HSC 2016 4U Marathon - Advanced Level All topics? Some ones that I can think of on the top of head that don't require any 4U knowledge: \\ $Prove that if and only if the sum of the digits of a number are divisible by 3, then the number is divisible by 3

Re: HSC 2016 4U Marathon - Advanced Level Well done

Re: HSC 2016 4U Marathon - Advanced Level \\ a,b,c \ $are integers so that$ \ a^2 + b^2 = c^2 \ $and$ \ a < b < c \\ $Prove that$ \ a + b + c \ $is always even$

Answered here: http://community.boredofstudies.org/13/mathematics-extension-1/331695/hsc-2015-3u-marathon-48.html#post7057100 ------------------------------------ \\ $i) Using the definition$ \ \binom{n}{r} := \frac{n!}{(n-r)!r!} \\ $So,$ \ \binom{n}{r} = \frac{n!}{(n-r)! r!} = \frac{n!}{(n-...