HSC 2016 MX2 Marathon ADVANCED (archive) (3 Viewers)

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seanieg89

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Re: HSC 2016 4U Marathon - Advanced Level

Written as intended, in step 2 you are only dealing with finite sums and integration is not the only thing you are doing in step 4. I just didn't want to be too explicit/prescriptive about the limiting argument, as methods will vary depending on level of mathematical knowledge.

If you naively allow exchanges of infinite series and integrals, then this calculation gives you zeta(2) quite speedily. If any university students attempt this, I would hope for a little more care.
 

seanieg89

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Re: HSC 2016 4U Marathon - Advanced Level

And a more algebraic question:

 
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Sy123

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Re: HSC 2016 4U Marathon - Advanced Level

 

seanieg89

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Re: HSC 2016 4U Marathon - Advanced Level

Good stuff :). (Although the denominator should be d+1 in your final answer to 4. Similarly, you have lost a constant factor on your way to the final expression in 3, although of course this does not affect the proof.)

Ps, the expression you obtained at the end of 3 is basically what you were looking for in 4, so you really didn't need to do that calculation again in 4. My intention for 3 was simply to note that any d-th degree polynomial in n (eg n^d) can be written as a linear combination of the polynomials P_k(n) for k =< d, which from the previous part can individually be written as a difference that telescopes.
 

Sy123

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Re: HSC 2016 4U Marathon - Advanced Level

Should be simple, but it does not fit in the regular marathon:

 

Paradoxica

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Re: HSC 2016 4U Marathon - Advanced Level

Should be simple, but it does not fit in the regular marathon:

For anyone attempting this, it is sufficient to do the problem for



as the pure integer part is incommensurable with the irrational parts. (recall the definition of an irrational number)
 

Sy123

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Re: HSC 2016 4U Marathon - Advanced Level

For anyone attempting this, it is sufficient to do the problem for



as the pure integer part is incommensurable with the irrational parts. (recall the definition of an irrational number)
This defeats the purpose of the question, if someone wants to do it this way, they'll need to prove the facts that they'd need to, to use it, along the way.
 

Paradoxica

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Re: HSC 2016 4U Marathon - Advanced Level

This defeats the purpose of the question, if someone wants to do it this way, they'll need to prove the facts that they'd need to, to use it, along the way.
what facts? the separability of the equation is obvious.

also, questions have a purpose now?

you didn't restrict the means of resolving this question.

and doing so defeats the purpose of this whole thread.
 
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Paradoxica

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Re: HSC 2016 4U Marathon - Advanced Level

This defeats the purpose of the question, if someone wants to do it this way, they'll need to prove the facts that they'd need to, to use it, along the way.
In any case, the question is still easy, with or without seperation of equations.

The equation rearranges by inspection into:



provided none of a or b is zero (see, told you seperation is obvious)

This would, as a consequence make √6 rational, which is false. Therefore, there does not exist a solution in the case of non-zero a,b

Then a and b are 0 forcing c to be 0.

Now go eat a snickers or something.
 

seanieg89

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Re: HSC 2016 4U Marathon - Advanced Level

He isn't "restricting the means" of resolving the question, he is just saying that a solution that jumps in halfway through after an unjustified reduction of the problem is not a complete solution. (And I agree with him.)

I don't think the leap from the question to your reduction is one that high school students should make without careful justification, as I think many of them would either be unable to see why this leap is true or worse, hoodwink themselves into believing that they understand the reduction when they don't.
 
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