HSC 2016 MX2 Marathon ADVANCED (archive) (1 Viewer)

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Sy123

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Post questions within the scope of Mathematics Extension 2 that are in general Q16 and beyond, focusing on problem solving and neat results within the reach of elementary mathematics.
 

InteGrand

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Re: HSC 2016 4U Marathon - Advanced Level

Post questions within the scope of Mathematics Extension 2 that are in general Q16 and beyond, focusing on problem solving and neat results within the reach of elementary mathematics.
I remember the 2014 HSC 4U Advanced marathon was allowed to continue for a couple of years. Why not the 2015 one (or why did the 2014 one have to stop)? (Though the 2015 one probably wasn't as good or vibrant as the 2014 one, so it wouldn't be as fun to keep it going I suppose.)
 

InteGrand

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Re: HSC 2016 4U Marathon - Advanced Level

Since 2016'ers probably have just started complex numbers recently, relatively easy Q to start off:

 

Sy123

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Re: HSC 2016 4U Marathon - Advanced Level

 
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Sy123

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Re: HSC 2016 4U Marathon - Advanced Level

I remember the 2014 HSC 4U Advanced marathon was allowed to continue for a couple of years. Why not the 2015 one (or why did the 2014 one have to stop)? (Though the 2015 one probably wasn't as good or vibrant as the 2014 one, so it wouldn't be as fun to keep it going I suppose.)
2014 (actually 2013) stopped because mods wanted it to (I wanted it to keep going)
 

InteGrand

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Re: HSC 2016 4U Marathon - Advanced Level

2014 (actually 2013) stopped because mods wanted it to (I wanted it to keep going)
Haha yeah, it was from 2013. And started in calendar year 2012 I think.
 

Drsoccerball

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Re: HSC 2016 4U Marathon - Advanced Level

Here's a crappy one I made :p





 

calamebe

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Re: HSC 2016 4U Marathon - Advanced Level

Since 2016'ers probably have just started complex numbers recently, relatively easy Q to start off:

Proof: http://m.imgur.com/QApReFi

I'm guessing the geometric interpretation is that if you subtract w from z, it will be 'longer' (larger modulus) or be equal to the length of subracting the modulus of w from the modulus from z. Though that was a complete guess.

Small error: http://m.imgur.com/glJoL07
 
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DatAtarLyfe

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Re: HSC 2016 4U Marathon - Advanced Level

Oh cool theres a 4u section now for us. Now i have something to do when i'm bored
 

Drsoccerball

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Re: HSC 2016 4U Marathon - Advanced Level

I think I'll recruit some of the new year 12s in my school to this forum.
 

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dan964

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Re: HSC 2016 4U Marathon - Advanced Level

Or in other terms the minimum distance between one point Z and another X on the two concentric circles i.e. |z-w| is the distance between the concentric circles itself ||z|-|w||. Equality occurs when arg z = arg w (it is a pointless statement if either z or w is 0)

It can simply be geometrically proved by using the triangle inequalities of the points O (origin), Z and W
|w|+|z-w|>= |z|
and |z|+|z-w|>= |w|

Rearranging gives
|z-w|>= |z|-|w|
and |z-w| >= |w|-|z|
which can leads to the generalised result:
|z-w|>= ||z|-|w||
 
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lita1000

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Re: HSC 2016 4U Marathon - Advanced Level

What do you mean by using Demoivres theorem? It's not exactly clear what you're asking for, unless you mean like use the demoivre's theorem to derive the double angle formula for cos, and use that to find the exact value of cos (5pi/12), which is the squared value of the expression in part 1
 

Zen2613

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Re: HSC 2016 4U Marathon - Advanced Level

What do you mean by using Demoivres theorem? It's not exactly clear what you're asking for, unless you mean like use the demoivre's theorem to derive the double angle formula for cos, and use that to find the exact value of cos (5pi/12), which is the squared value of the expression in part 1
I think he wants us to use demoivre to find an expression for cos(4theta) in terms of cos(theta) because that will be a quartic which will probably resemble his 16x^4 - 16x^2 -1 = 0 thingo.
 
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