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  1. Paradoxica

    The Evil Integral Marathon

    This is purely an observational thread, although you may choose to answer them anyway. Post the most evil integrals you -have ever/happen to- come across in your mathematical endeavours. And optionally, make a few observational comments on the integral, and any attempted evaluations of it...
  2. Paradoxica

    Finding conditions for monotone convergence of a particular sequence.

    Suppose f is a non-positive XOR non-negative, singularity free, smooth, C∞ function on the interval [0,1]. Define the following sequence of numbers, which approximate the integral of f on [0,1]: an = nth Left Riemann Sum with uniform partition bn = nth Right Riemann Sum with uniform partition...
  3. Paradoxica

    Logic Puzzles Marathon

    I'm surprised there wasn't one already. Rules as per other marathons, this one is for those problems that are more logical than mathematical. $\noindent The other $n-1$ statements are false$ \\ $The other $n-1$ statements are false$ \\ $The other $n-1$ statements are false$ \\ \vdots \\ \vdots...
  4. Paradoxica

    Extracurricular Elementary Mathematics Marathon

    Rules are as per the other marathons. Difficulty should be reasonable, and hints should be provided as deemed necessary. Use any "elementary" techniques, provided you state them, and if it's fairly advanced, outline a proof. I'll start off simple. $Find all integer pairs $(x,y)$ that satisfy...
  5. Paradoxica

    Insight on a particular class of curves.

    $\noindent$ x^x + y^y = k_1$; $x^y + y^x = k_2$; $x^{-x} + y^{-y} = k_3$; $x^{-y} + y^{-x} = k_4 \\$\noindent For $k_1 = k_2 \approx 1.3844012551107 $ and $ k_3 = k_4 \approx 2.8893357220195$, the second and fourth curves approach a vanishing point on the Cartesian plane. As these curves...
  6. Paradoxica

    Interesting mathematical statements

    $\noindent Leave an intriguing mathematical statement for all of us to be flabbergasted by.$ $\noindent I'll start.$ \lim_{x \rightarrow \infty} e^{e^{e^{\left ( x+e^{-(a + x + e^x + e^{e^x})} \right )}}} - e^{e^{e^x}} \equiv e^{-a} $\noindent For all values of "$a$".$
  7. Paradoxica

    Fascinating and stumping problem from Scientific American.

    You have a single blue cube and an unlimited number of red cubes, all of identical dimensions. What is the maximum number of red cubes that can touch the blue cube along it's sides or part of it's sides? Heavy discussion has occured for this problem at Brilliant.org, which is where I...
  8. Paradoxica

    Geeks And Dorks: The Webcomic Thread

    So, what are people's favourite webcomics here? I'll start with my own. Saturday Morning Breakfast Cereal (Zach Weiner) XKCD (Randall Munroe) Square Root of Minus Garfield (David Morgan-Mar and many others) Schizmatic: A Webcomic Of Intelligent Weirdness (Michael Yu) Irregular Webcomic (David...
  9. Paradoxica

    QQ on Max/Min

    If a part of a question says something like "find the maximum/minimum of R(x)" where R(x) = O(x) + P(x)/Q(x) (a.k.a. some rational function), is it acceptable to quickly prove AM-GM, and then proceed to use AM-GM to find the minimum/maximum value of the function? Of course, I am assuming there...
  10. Paradoxica

    Question regarding IBP

    Has there ever been a HSC question where the student was required to use Integration By Parts more than twice?
  11. Paradoxica

    Preparation for Year 11 and Year 12 Maths

    Recently, the teacher of the M1 class in my school loaned me one of the few existing copies (in the school) of "Revised 4 Unit Course - A Higher School Certificate Course in Mathematics Years 11 and 12". So this got me thinking. What should I be preparing ahead of time to make the path easier...