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  1. Paradoxica

    Same Sex Marriage Debate

    oh i'm not here to participate; i'm here to cast judgement on dan as he is casting on the entirety of the non-straight population.
  2. Paradoxica

    MX2 Integration Marathon 2021

    you don't?
  3. Paradoxica

    Same Sex Marriage Debate

    it is clear that dan has a naïve or possibly non-existent understanding of the psychology of sexuality (and even if it exists, it is heavily clouded by religious puritanicalism), so i would take anything he says about it with a molecule of salt.
  4. Paradoxica

    Advanced mathematics

    obviously they have access to a hyperbolic time chamber
  5. Paradoxica

    What is your iq?

  6. Paradoxica


    Personally I don't see a problem with this, but you would have to prove that a for f a strictly increasing function, f applied to a strictly increasing sequence is a strictly increasing sequence. However, I feel like this lemma has been stated without proof in past HSC questions, so I don't know.
  7. Paradoxica


    Random observation I want to make. Let a = \tan{A}, b = \tan{B}, c = \tan{C} Then the problem reduces to maximising \cos(2A) + \cos(2B) + 2\cos C subject to the constraint \cos(A+B+C) = 0 and A,B,C are acute angles. I don't know if this has a nice geometric interpretation, but if anyone...
  8. Paradoxica

    Proof by induction question. It's been rotting my brain. Question 50

    actually there is a systematic way of doing this and it's basically the discrete version of the derivative Define the forward difference operator Δf(n) = f(n+1)-f(n) the binomial coefficients xCk are the monomial analogues of the continuous monomials x^k /k! in the sense that Δ(xCk) =...
  9. Paradoxica

    Proof by induction question. It's been rotting my brain. Question 50

    use the greedy algorithm to successively eliminate the highest available power by subtracting the appropriately weighted binomial coefficient this is a lot of expanding and algebra simplification so it's not that much nicer than splitting into cases and arguing carefully, but it doesn't require...
  10. Paradoxica

    Permutation and combination question (pretty tough)

    yeah and we didn't sugar coat it by splitting into odd and even cases in the main steps of the proof :biggrin:
  11. Paradoxica

    Proof by induction question. It's been rotting my brain. Question 50

    Black Magic Proof: Observe the following identity: n^5+n^3+2n \equiv 120\binom{n}{5} + 240\binom{n}{4} + 156\binom{n}{3} + 36\binom{n}{2} + 4\binom{n}{1} All coefficients of the binomial sum are multiples of 4, and all binomial coefficients are integers, so the RHS is divisible by 4, hence...
  12. Paradoxica

    Is chess a sport?

    everything that has been posted here is irrelevant the aspect of chess that makes it sport-like is the fandom and the audience/culture that is created around the elite players commentators, supporters, etc. just like any other e-sport
  13. Paradoxica

    Who will be looking for love at Uni?

    don't try it's hopeless :biggrin: on the other hand you never know unless you shoot your shot
  14. Paradoxica

    complex number help!!

    Here is a decidedly slicker way to decide which root is α (IMO) in this image, α = t
  15. Paradoxica

    complex number help!!

    Consider the trigonometric form of α-β by rewriting it in terms of powers of ρ
  16. Paradoxica

    MX2 Integration Marathon 2021

    the real answer to this conundrum is that by using a substitution which maps a finite open interval to ℝ, you cannot invert the substitution to obtain an antiderivative valid over all of ℝ. the substitution is only bijective locally, not globally, so a true inverse does not exist, hence the...
  17. Paradoxica

    MX2 Integration Marathon 2021

    Define \psi(a,k,u,v) := \log_a\left(a^{u} + k + a^{v}\right) The original question takes the argument a=2. For brevity, we will suppress the first two arguments⁰, as it turns out they are irrelevant to the final answer. (This is a thing in higher mathematics lmao get used to it) The...
  18. Paradoxica

    HSC Extension 1 Mathematics Predictions / Thoughts

    Not everyone has such generous life opportunities though. Some of us grew up just trying to survive in a hostile home AND school environment, parents who don't care or understand, an unreconcilable communication barrier with parents, and a plethora of neuro/psych impairments. Mathematics was my...
  19. Paradoxica

    HSC Extension 1 Mathematics Predictions / Thoughts

    *looks at the world* you're out of your mind
  20. Paradoxica

    Hard Proofs Question

    So I solved (v) with two applications of the Stolz–Cesàro theorem, I have given up on attempting a more within-the-syllabus proof.