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Re: HSC 2014 4U Marathon - Advanced Level \\ $Let$ \ H_n = \sum_{k=1}^n \frac{1}{k} \\ \\ $Find$ \ \sum_{n=2}^{\infty} \frac{1}{nH_n H_{n-1}}

Re: HSC 2014 4U Marathon Ah I see, ok, my bad

Re: HSC 2014 4U Marathon What you are talking about is the orthocenter, but the orthocenter is not the centroid of a triangle and is not equidistant to all 3 points. Here is a hint: The general equation of a circle in the co-ordinate plane is: (x-h)^2 + (x-k)^2 = r^2

Re: MX2 Integration Marathon \int_0^{\pi /4} \frac{1}{\cos x \sqrt{\cos 2x}} \ dx

Re: HSC 2013 3U Marathon Thread Yes well done --- \\ $A number is divisible by 3 if the sum of its digits are divisible by 3$ \\ \\ $Find how many 4 digit numbers are even and divisible by 3$

Re: HSC 2014 2U Marathon \\ $i) Solve the system of equations$ \\ \\ x+y+z = 0 \\ 5x + 4y + 3z = 0 \\ 6x + 3y + 2z = 1 \\ \\ $ii) Hence find constants$ \ A,B,C \ $so that$ \\ \\ \frac{1}{(x+1)(x+2)(x+3)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{x+3} \\ \\ $iii) Hence find$ \ \ \int_{-1}^1...

Re: HSC 2014 2U Marathon Yes it is 2U

Re: HSC 2014 4U Marathon - Advanced Level Yes well done \\ $A base 2 representation of a number is using only 2 digits$ \ 1 \ $and$ \ 0 \ $to represent it, we commonly use decimal form, or base 10$ \\ \\ $That is, in normal numbers we have$ \ 123 = 1 \times 10^2 + 2 \times 10^1 + 3 \times...

Re: MX2 Integration Marathon Yes well done \\ \int_0^{\pi/4} \frac{1}{1+\sin^2x} \ dx

Re: HSC 2014 4U Marathon \\ $Explain why there is only 1 circle that will pass through 3 non-collinear points$

Re: HSC 2014 2U Marathon \\ $i) If both the trapezium rule and Simpson's rule were used to estimate the area under a linear function, which would give more accurate results?$ \\ \\ $ii) What if the function was a parabola?$ (If you're a 4U student, please refrain from answering questions in...

Re: HSC 2013 3U Marathon Thread \\ $Prove$ \ \sum_{k=1}^n \frac{(-1)^{k-1}}{k+1}\binom{n}{k} = \frac{n}{n+1}

Re: HSC 2014 2U Marathon \\ $i) Find$ \ \int_0^4 \frac{x}{x^2+1} \ dx \ \ \ \fbox{2} \\ \\ $ii) Using the trapezium rule on the integral in part (i), estimate the value of$ \ \ln (17) \ \ \ \fbox{3}

Re: HSC 2014 4U Marathon - Advanced Level \\ $The angles of a triangle$ \ \alpha, \beta, \gamma \ $are opposite sides$ \ a,b,c \ $respectively$ \\ \\ $This triangle is such that$ \ a^2+b^2 = kc^2 \ $for some constant$ \ k>1 \\ \\ $Find$ \ \frac{\cot \gamma}{\cot \alpha + \cot \beta}

Re: MX2 Integration Marathon \\ $Find$ \ \lim_{t \to \infty} \int_0^t \frac{|x-1|}{(x+1)(x^2+1)} \ dx

Re: HSC 2014 4U Marathon \\ $Given$ \ \sec x + \tan x = A \ $for some non-zero real$ \ A \ $find$ \ \csc x + \cot x \ $in terms of$ \ A

Re: HSC 2014 4U Marathon - Advanced Level \\ $Re-arrange, the problem now becomes to prove$ \\ \\ n\sqrt[n]{2} < \frac{n+1}{n} + \frac{n+2}{n+1} + \dots + \frac{2n}{2n-1} \\ $Using the general AM-GM inequality, we get$ \\ \\ \sum_{k=0}^{n-1} \frac{n+k+1}{n+k} \geq n\sqrt[n]{\frac{n+1}{n}...

Have a look at the Advanced level marathon, a lot of reasonably hard inequality questions are there, and although they may use some slightly more advanced techniques (i.e. General AM-GM, Cauchy and Jensen's inequality), the skill set required translates well into the HSC environment. A lot of...

Re: HSC 2014 4U Marathon - Advanced Level Ah yes I meant continuous And yes you got it (can you please post a problem)

Done Also what was the point of those last 2 questions? (just curious)