HSC 2014 MX2 Marathon ADVANCED (archive) (2 Viewers)

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seanieg89

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Re: HSC 2014 4U Marathon - Advanced Level

Yep, I figured that Sy was going for this approach of reduction formula --> factorials --> substitute in value.

Now that we've 'guessed' what the value of (1/2) factorial is, I think it'd be good to have a follow up question actually proving that (1/2)! = root(pi)/2 using the Gamma function. Might be a bit difficult (or even impossible) within MX2 constraints though since the reduction formulae used for the integral x^{n-1}*e^{-x} from 0 to 1 is not considered to be continuous.
Well, evaluating Gamma(3/2) is the same integral as the Gaussian. Just from 0 to infinity, and with a possible constant factor floating around somewhere. This can be done (not rigorously of course, but as rigorous as the mx2 course gets.)

I don't think there is any mx2-able method to prove the gamma-beta relation which would be an alternate method of proof.
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

 
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seanieg89

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Re: HSC 2014 4U Marathon - Advanced Level

i) a = b = 2N is an example, for each N.

ii) Are there meant to be additional assumptions or something?

1/3 = 1/6 + 1/6 = 1/4 + 1/12 is a counter-example to the claim.
 

RealiseNothing

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Re: HSC 2014 4U Marathon - Advanced Level

Without using the AM-GM for 3 real numbers, show that:

 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

i) a = b = 2N is an example, for each N.

ii) Are there meant to be additional assumptions or something?

1/3 = 1/6 + 1/6 = 1/4 + 1/12 is a counter-example to the claim.
I apologize throughout the entire question a, b are distinct.
 

Ikki

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Re: HSC 2014 4U Marathon - Advanced Level

What does that pi looking thing mean? (Noob alert hahaha)
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

What does that pi looking thing mean? (Noob alert hahaha)
It is the same thing as but for products =)

i.e.
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

oops
 

Sy123

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Re: HSC 2014 4U Marathon - Advanced Level

Attempt #2



 

seanieg89

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Re: HSC 2014 4U Marathon - Advanced Level

Attempt #2



There is a bit of a logical flaw here, remember that we are trying to show that NO polynomial of the prescribed form has all roots positive and real.

You have just provided one example of a polynomial of the prescribed form that does not have all roots positive (any poly that is of the prescribed form and additionally has a_0=0), this is not what the question is asking for.


You are on the right track dealing with inequalities and symmetric sums, seeking a contradiction.
 

braintic

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Re: HSC 2014 4U Marathon - Advanced Level

Surely its zero? Or better: 0+ if m>n, 0- if m<n

Not sure how to show working though (at least not without making the same sort of assumptions I have made in my head already).
 
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