2012 Year 9 &10 Mathematics Marathon (2 Viewers)

russ3l

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Re: 2012 Year 9 &10 Mathematics Marathon

We're the only year 10s, but meh, who cares. To be honest, I reckon russ3l and ymcaec should come tomorrow. It would be interesting. :)
I would've loved to come to this 'meat' thingy ma bob :) but ive gotta get my swimming up, sorry :p Perhaps we can add eachother on facebook so we 'meet' eachother :tongue2:
 
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Re: 2012 Year 9 &10 Mathematics Marathon

I would've loved to come to this 'meat' thingy ma bob :) but ive gotta get my swimming up, sorry :p Perhaps we can add eachother on facebook so we 'meet' eachother :tongue2:
I don't think Demento has facebook yet :p
 

RealiseNothing

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Re: 2012 Year 9 &10 Mathematics Marathon

Let be an integer. We are given a balance and weights of weight . We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.



This is a cool arrangements question.

WARNING: EXTREMELY DIFFICULT.
 
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Re: 2012 Year 9 &10 Mathematics Marathon

Well for i) Probability of winning first dart = p

Probability of winning with second dart given first dart did not hit

If you expand that you should get:


For ii) To win, she needs to hit the target twice, there are 4 possibilities:

There are 3 possibilities of hitting 2 darts with one missing (i.e. 1st and 2nd, 2nd and 3rd, 1st and 3rd).

The probability of this is:

1 permutation of hitting with all 3 darts:

So the total probability


For iii) I hope I didn't type anything wrong...




If Darcy is to be more likely to win game 1 than game 2, then from above:





Since







Therefore for 0 < p < 1, this statement is true and the induction holds.


For iv), algebra can really finally be used to some extent.



Divide by p, get it into the format of and use the quadratic formula:





Thus, you have a quadratic equation to solve and the quadratic formula is used which I cannot bother with typing unfortunately...

Eventually you get p = 1.39, 0.36. Only one of these values is between our limit 0 < p < 1, so the answer would be when p = 0.36 (2 dp).
Yes, the final answer is exactly

Congratulations! You have successfully completed (the last part of) Question 6 of the 2011 HSC Mathematics Extension 1 Paper

:)
 

RealiseNothing

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Re: 2012 Year 9 &10 Mathematics Marathon

People post up hard questions just to troll us year 10s -.-
No, hard questions (like the one I posted) get you thinking and allow you to explore and develop new methods and ideas.
 

russ3l

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Re: 2012 Year 9 &10 Mathematics Marathon

Yes, the final answer is exactly

Congratulations! You have successfully completed Question 6 (the last part may I add) of the 2011 HSC Mathematics Extension 1 Paper

:)
-___________-
 

iBibah

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Re: 2012 Year 9 &10 Mathematics Marathon

Let be an integer. We are given a balance and weights of weight . We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.



This is a cool arrangements question.

WARNING: EXTREMELY DIFFICULT.
Yes I remember this. Great question.
 

ymcaec

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Re: 2012 Year 9 &10 Mathematics Marathon

Yes, the final answer is exactly

Congratulations! You have successfully completed (the last part of) Question 6 of the 2011 HSC Mathematics Extension 1 Paper

:)
so the hsc ext 1 is not that hard ... :caffeine:
 

Demento1

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Re: 2012 Year 9 &10 Mathematics Marathon

Let be an integer. We are given a balance and weights of weight . We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.



This is a cool arrangements question.

WARNING: EXTREMELY DIFFICULT.
Going to take a read and go to sleep thinking about it. If I do not have an answer by morning, I may have to ask you for a solution. Apologies, mathematics isn't my best subject. :(

Yes, the final answer is exactly

Congratulations! You have successfully completed (the last part of) Question 6 of the 2011 HSC Mathematics Extension 1 Paper

:)
Let's hope I get Q6 in 2014 HSC 3U maths right. :p
 

ymcaec

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Re: 2012 Year 9 &10 Mathematics Marathon

Let be an integer. We are given a balance and weights of weight . We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.



This is a cool arrangements question.

WARNING: EXTREMELY DIFFICULT.
i hope these kinds of question only exist in competitions ... not hsc...
P.S. i got my answer from my friend ... Mr. (or Ms.) Google :)
 

ymcaec

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Re: 2012 Year 9 &10 Mathematics Marathon

Relax, that question can be solved using year 10 probability methods :)



Haha, that's just a small portion of it. Although I'm sure you'll do very well in it (and MX2 as well!)
i hope the questions like the weights one above is not in the hsc :)
 

iBibah

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Re: 2012 Year 9 &10 Mathematics Marathon

Going to take a read and go to sleep thinking about it. If I do not have an answer by morning, I may have to ask you for a solution. Apologies, mathematics isn't my best subject. :(



Let's hope I get Q6 in 2014 HSC 3U maths right. :p
Not to worry because there is no shame in not being able to do this one, especially a year 10. Though you did quite well with that ext 1 question so I'm sure you have potential in maths.
 

iBibah

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Re: 2012 Year 9 &10 Mathematics Marathon

i hope the questions like the weights one above is not in the hsc :)
Well let's just say IF you can do it, the HSC will be no problem for you.

It looks so difficult when you read it. Before attempting it, just try and understand the criteria more by testing n=0,1,2,3 just to get an idea of where you are heading before you attempt to solve it.
 

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