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Re: HSC 2016 4U Marathon - Advanced Level The extension I mentioned earlier: Can you show that this polynomial is the polynomial of least degree passing through these n points? I think you most likely need some slightly out of syllabus stuff for this though, so don't spend too much time on it.

Re: HSC 2016 4U Marathon - Advanced Level Yep exactly. It's a better question if you have not come across interpolation before.

Re: HSC 2016 4U Marathon - Advanced Level That is exactly what he said, in different words. I don't think there is any ambiguity in his post. He didn't say anything about injectivity. Saying abscissas are different is just saying that no two points in this set have the same x-coordinate...

Re: HSC 2016 4U Marathon - Advanced Level I don't think that is what he is asking. He is asking for an explicit expression for a polynomial whose graph passes through a given finite set of points in the plane. (An extension question for those with knowledge from outside syllabus is to show...

Re: MX2 2016 Integration Marathon J=\int_0^\infty \frac{x\log{x}}{(x^2+a^2)(a^2x^2+1)}\, dx\\ \\ =\int_\infty^0 \frac{\frac{1}{x}\log{\frac{1}{x}}}{\left(\frac{1}{x^2}+a^2\right)\left(\frac{a^2}{x^2}+1\right)}\, \frac{-dx}{x^2}\\ \\ \\ =-\int_0^\infty \frac{x\log{x}}{(x^2+a^2)(a^2x^2+1)}\...

Which part is confusing you? This is basically just the definition of differentiability. Remember that we say that f(x) is differentiable at a if \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h} exists. If it does, we denote this quantity by f'(a) which will just be a real number depending on a...

No, singularities are not allowed, because the function is continuous and has domain R. (We cannot have any domain "holes".)

Composition cannot create parity? Why not? Eg f(x)=1-x is not odd or even, but composed with itself is the identity which is odd. It is true that the only polynomial function that works is the identity and yes degree considerations give you this quickly as you note. This isn't too...

Yeah, you have done something fishy in deducing that f=f^{-1}, care to explain your reasoning if you still believe this fact after thinking more? Note also that you have used continuity nowhere. This is essential, as we have a vast array of solutions to the functional equation if continuity is...

Find with proof, all continuous functions f:R->R such that: f(f(f(x)))=x for all real x.

Oh yes, entirely. That was why I considered it cute, I quite like simple ideas exploiting symmetries like that.

Yep, you got the key idea, just a silly error. Was what intentional?

I do not believe your first solution is correct, neither is the statement that if f(x) is a solution then f(2x/(3x-2)) is one too. This would be the case if the functional equation was symmetric in these two unknowns, but the factor of 1/2 stops that from happening. A quick way to verify that...

Here's a cute one: Find all solutions f: R\{2/3}->R to: 504x-f(x)=f(2x/(3x-2))/2.

I don't know how easy/natural it would be to make such a geometric argument rigorous, as this seems to be more of an algebraic/number theoretic fact. Like, you can tesselate the complex plane by any kind of rhombus you like (with the base one having vertices 0,1,1+w,w where w is an arbitrary...

By this I assume you are asking why the only rings of the form S={a+bw : a,b integers} (equipped with the standard sum and product operations of C) where w is a complex number of unit modulus are the Gaussian integers and the Eisenstein integers. (If you mean something else please clarify)...

Also, there are only finitely many solutions to that functional equation I believe. It is clear that if a solution vanishes somewhere it vanishes everywhere (let y be a root to make the RHS vanish and then vary x so the LHS vanishing tells you that f is identically zero). Let us assume now...

I could. I could also say: "Justify your answer by way of proof", or countless variations of this. I made a choice, and the meaning was unambiguous. You will be frequently frustrated in future studies if such variations in wording bother you.

Condescending much? Let people choose their own words. Clarity and precision of semantic content is far more important in mathematical writing than such choices in wording. Especially in a forum post of a single question lol. I did not write this question as "Prove X" simply because I wanted...

Bingo :), back yourself more! Will let someone else post one now.