HSC 2013 MX2 Marathon (archive) (1 Viewer)

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Sy123

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Re: HSC 2014 4U Marathon

By lowest gap I'm assuming you mean smallest gap? I don't think is the smallest gap. Consider:



would be the smallest gap between all number of the form where the denominator is fixed and numerator varies. But the denominator isn't necessarily fixed.
The smallest gap yes, I mean that the smallest gap between any 2 rational numbers is 1/(common denominator).

Take arbitrary rationals with integer variables





In order to minimize the gap between two arbitrary numbers




But using the p,q,r notation is easier I think it says the same thing however



Meaning the middle bound is still rational
 

ganji

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Re: HSC 2014 4U Marathon

Show that the points z_1,z_2,z_3 are collinear if and only if αz_1+βz_2+γz_3=0, where α+β+γ=0 (α,β,γ are real and non zero)

P.s. How do you use Latex here?
 

RealiseNothing

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Re: HSC 2014 4U Marathon

The smallest gap yes, I mean that the smallest gap between any 2 rational numbers is 1/(common denominator).

Take arbitrary rationals with integer variables





In order to minimize the gap between two arbitrary numbers




But using the p,q,r notation is easier I think it says the same thing however



Meaning the middle bound is still rational
Ok I see what you mean. Using p, q, r notation though there is no guarantee that and immediately follow each other. So I think instead of using the smallest gap as I think you should let the two rational numbers be say and and assume that they immediately follow each other (that is there is no rational number between A and B). From here the smallest gap is by assumption, which can be used in your inequality to get instead:



Where is some irrational number greater than 1, and is rational as it is the difference of two rational numbers.
 

Sy123

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Re: HSC 2014 4U Marathon

This is a good inequality:



 

seanieg89

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Re: HSC 2014 4U Marathon

Wow haha, my way is much much longer but uses more basic inequalities
Haha the gap in lengths would probably be at least partially closed by me having to prove Jensen's before using it. I will post a statement and proof of it here later today, just out of interest to see how compact and elementary I can make it.
 

RealiseNothing

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Re: HSC 2014 4U Marathon

Show that diverges to infinity.

Note the angle is in radians.
 

Sy123

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Re: HSC 2014 4U Marathon

This is a good inequality:



Here is my long solution;









So yeaa this might be longer than your proof including the proof of Jensen's inequality
 
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seanieg89

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Re: HSC 2014 4U Marathon

Here is my long solution;









So yeaa this might be longer than your proof including the proof of Jensen's inequality
Nice!

Yeah imo Jensen's is extremely powerful for how easy it is to prove.
 

RealiseNothing

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Re: HSC 2014 4U Marathon

You gave your sum of a_k to go from 1 to n, but your product makes no mention of n.

I presume you want the limit as n-> infinity, in which case the a_k series you provided converges to 1?
Yep that's it, I initially wanted the question to be in terms of 'n' then changed to letting n -> infinity but forgot to change the sum of a_k.
 

Sy123

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Re: HSC 2014 4U Marathon



AM-GM will lead to an inequality that is not tight enough to cover the bound in question
 
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