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HSC 2013 MX2 Marathon (archive) (7 Viewers)

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Sy123

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Re: HSC 2013 4U Marathon

Great question:

First lets establish the identity



Take the natural logarithm of both sides, ensuring that cos(x/2^k) is always positive (absolute value), and use logarithm laws



Differentiate both sides



Take r to infinity











 

Lieutenant_21

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Re: HSC 2013 4U Marathon

Great question:

First lets establish the identity



Take the natural logarithm of both sides, ensuring that cos(x/2^k) is always positive (absolute value), and use logarithm laws



Differentiate both sides



Take r to infinity











Great solution! There is a little shorter way:















================================================================================

Have a go at this integral:

 

seanieg89

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Re: HSC 2013 4U Marathon

Pretty sure that integral can't be expressed in terms of elementary functions.
 

seanieg89

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Re: HSC 2013 4U Marathon

Arctan and cos are both elementary functions themselves lol. I still don't think it can be done though.
 

seanieg89

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Re: HSC 2013 4U Marathon

Yes, I know arctan and cos are elementary functions but you need poly-logarithm functions to do it.
Yeah I know, I was just responding to you saying you think arctan(cos x) can't be expressed in terms of elementary functions.

Anyway, all good :).
 

Lieutenant_21

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Re: HSC 2013 4U Marathon

From the early 70s. Those problems are often doable by good mx2 students with no prior olympiad training and this one is no exception.
Could you please elaborate on that? :)
What do people do to prepare for Olympiads? They clearly need excellent problem solving skills which is something I want to improve.
 

seanieg89

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Re: HSC 2013 4U Marathon

Solve problems haha. And you can find sort of unofficial IMO syllabi various places online to get an idea of useful pieces of theory that are still considered to be at a high school level. (As opposed to advanced university uses of calculus, which are not tested as calculus is not taught properly in high school.)

The point is more to answer things very rigorously with elementary methods. There are many books/articles on problem solving strategies out there, but honestly practicing and reviewing your though processes is probably the most beneficial "training".
 

Lieutenant_21

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Re: HSC 2013 4U Marathon

Solve problems haha. And you can find sort of unofficial IMO syllabi various places online to get an idea of useful pieces of theory that are still considered to be at a high school level. (As opposed to advanced university uses of calculus, which are not tested as calculus is not taught properly in high school.)

The point is more to answer things very rigorously with elementary methods. There are many books/articles on problem solving strategies out there, but honestly practicing and reviewing your though processes is probably the most beneficial "training".
Thank you!
So I will just keep doing questions.
 

hayabusaboston

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Re: HSC 2013 4U Marathon

Assume it is rational, so can be expressed as a fraction where the numerator and denominator have no common factors:







So now we know that is even and hence is even. So now we let :







So now we know that is even, and so is even.

However if both and are even, then they have a common factor of . This is a contradiction and so our assumption is wrong, that is, is not rational - it is irrational.
Is this not directly quoted from cambridge?
 

hayabusaboston

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Re: HSC 2013 4U Marathon

I am pretty sure that was the only mark I lost (hopefully, as I haven't gotten the whole paper back yet). The test wan't really that hard, was very similar to questions found in Cambridge.
I never used Cambridge before but I used it to study for this test because my teacher uses it and I think Cambridge and Terry Lee complement each other very well for 4U.
According to a lady from Dymocks on george street (Is it george street? lol I think so), cambridge 4u is "for students who are really struggling to grasp the concepts of extension 2 maths, and need help at a basic level"

Lol.

If ur test similar to cambridge, must have been friggin easy hahaha.
 
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