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HSC 2013 MX2 Marathon (archive) (4 Viewers)

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Sy123

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Re: HSC 2013 4U Marathon









 
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Re: HSC 2013 4U Marathon

Think you meant:

prove that x is a Fibonacci number if and only if 5x^2+4 or 5x^2-4 is a perfect square. (just being picky).
 

Sy123

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Re: HSC 2013 4U Marathon

Think you meant:

prove that x is a Fibonacci number if and only if 5x^2+4 or 5x^2-4 is a perfect square. (just being picky).
Yeah that sounds more correct, thank you
 
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Re: HSC 2013 4U Marathon

Think of the subsets of as picking elements from the set. That is, if you choose 0 elements, you have 1 choice. If you choose 1 element, you have n choices, for 2, nC2. Ie: which is clearer from your point of view from the expansion of On replacement of n by 2n we get the result.
 

Sy123

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Re: HSC 2013 4U Marathon













EDIT 1: One sec, I may have made a mistake
EDIT 2: It should be fine now
EDIT 3: Changed some restrictions, I hope its fine now (B < 1, A > 1)
 
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rural juror

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Re: HSC 2013 4U Marathon

We know that:



Now let and





Multiply both sides by 3 to establish the inequality:



Now consider:







Split the RHS up into the following:







Now using the inequality we establish before, we get:



Let this be A:



Hence since







i might have a quicker way and easier to find,

looking at x^6(y^-2 - x^-2) > y^6(y^-2 - x^-2) *just assume the greater than sign is greater than or equal to, this inequality can be easily shown for x>y and y>x to hold true, then all you have to do is move the fractions over and you have the result. max 3 lines or so
 

rural juror

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Re: HSC 2013 4U Marathon

oh same solution as sys thatll teach me to scroll down
 

Sy123

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Re: HSC 2013 4U Marathon

First post, let's see if I can get the latex right -- I'm studying how you've done it lol












Are you sure this one is correct? I'm fairly certain I can prove it has a lower bound of 9 and an upper bound of 90 pretty easily...
The first answer is correct, as for the second one, let me check my proof again, I got the result from 'Art of Problem SOlving' and it asked whether it diverged and converged.

Feel free to post your proof of the upper and lower bounds though

EDIT: Yeah my proof is wrong, I didn't consider the numbers where a zero is in the middle -.-
 
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Sy123

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Re: HSC 2013 4U Marathon

Another good Putnam question







 
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fionarykim

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Re: HSC 2013 4U Marathon

How are u guys typing so much maths here LOOOL im amazed

Also the questions just seem to just confuse my head, too hard ):
 
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