Spoken like a true Cambridge mathmo.(Original post by TheMagicMan)
I believe a necessary and sufficient condition would be something along the lines of having a measure 0 set of discontinuities on the domain of y in the solution.

und
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ukdragon37
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 10062013 00:02
(Original post by TheMagicMan)
Difficult. Didn't do as much work as I probably should have in the first two terms but I somewhat pulled it together for the exams.
(Original post by Jkn)
Well the question must have a solution by nature because either you can consistently predict the coin is counterfeit or you cannot
(Original post by Jkn)
...
EDIT: I.e. what I'm saying is answering "yes" to whether the method exists is not inherently more preferable to answering "no", and so there is no reason why you should prefer to say air resistance exists over it not existing as this is an answerinfluencing assumption.Last edited by ukdragon37; 10062013 at 00:10. 
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 10062013 00:04
(Original post by und)
Spoken like a true Cambridge mathmo. 
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 10062013 00:06
(Original post by und)
Spoken like a true Cambridge mathmo. 
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 10062013 00:10
(Original post by ukdragon37)
Glad to hear you did well though.
Telling that to a modern logician is like saying to a quantum physicist "Newtonian physics applies everywhere, including the atomic level."
You haven't actually answered my problem. Recall that your question asked "Whether a method exists", not "How will such a method work". The assumption that air resistance is not negligible is just as nonnegligible as the assuming the opposite, since it predisposes the answer. Hence you cannot say that an answer which utilises the air resistance assumption, in either direction by itself, is valid. Which is why I said unless you make it clearer then the only solution would be to give you the document listing all the possible assumptions and solutions.
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 10062013 00:11
(Original post by TheMagicMan)
Spoiler:ShowI haven't posted on here in ages, but my first offering will be the following, which is a slight adaptation of a question from the cambridge entrance exam in 1981 (so roughly STEP level)...
How many fifth powers are there of form ?
It requires AS knowledge at the most, and I know of no solution that uses more complex methods.
Hint:
Take , it is obvious that , so and note that . Hence, it is sufficient to take , then . 
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 10062013 00:12
(Original post by bananarama2)
Honestly. It's good that you arent in year three. I couldn't put up with this every time you two met
Posted from TSR MobileLast edited by ukdragon37; 10062013 at 00:13. 
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 10062013 00:15
(Original post by Mladenov)
It clearly has infinitely many solutions.
Take , it is obvious that , so and note that . Hence, it is sufficient to take , then .
How many 6th powers are there of form ? is more what I was aiming for. 
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 10062013 00:36
(Original post by TheMagicMan)
I didn't realise that 2013 was divisible by 11, which kind of destroys it. I was aiming to incorporate 2013 in my adaptation.
How many 6th powers are there of form ? is more what I was aiming for.
To clarify,Last edited by Mladenov; 10062013 at 00:39. 
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 10062013 00:40
(Original post by Mladenov)
x
(Original post by ukdragon37)
Telling that to a modern logician is like saying to a quantum physicist "Newtonian physics applies everywhere, including the atomic level."
You haven't actually answered my problem. Recall that your question asked "Whether a method exists", not "How will such a method work". The assumption that air resistance is not negligible is just as nonnegligible as the assuming the opposite, since it predisposes the answer. Hence you cannot say that an answer which utilises the air resistance assumption, in either direction by itself, is valid. Which is why I said unless you make it clearer then the only solution would be to give you the document listing all the possible assumptions and solutions.
EDIT: I.e. what I'm saying is answering "yes" to whether the method exists is not inherently more preferable to answering "no", and so there is no reason why you should prefer to say air resistance exists over it not existing as this is an answerinfluencing assumption.
Proving that air resistance is nonnegligible is the objective of the first part of the problem (to allow us to consider the 'drops' as analogous to 'scales'). We prove it is nonnegligible by first assuming assuming all of the law of Physics and then approximating each law as is appropriate for the problem. We decide what is appropriate by the situation; if we wanted to we could use all of the compacted laws, it would just take longer and mean we must seek further information by going out and doing experiments or modelling the coins and the bag and certain objects whilst remarking the validity of this assumption. Is Physics, this is the way every problem must be done and you are certainly correct in saying that this method would require a document listing all assumptions along with experimental evidence as to their validity. When we do Physics, we do not write such documents because we call on experiments performs in similar situations to tell us things about what might happen in our current situations. We establish mathematical relationships between the variables recorded in an experiment and this is how we get our Physical laws. Note that all laws must be accompanied with an analysis of how well this correlates with experiment as well as when it is valid to apply these laws. These documents make up the body of Physics and are the very documents to which you refer.
Whilst, in mathematics, we happily build our knowledge on top of the axioms. We also don't require pages of proofs for every theorem we use in every other proof we use them for. In Physics, whilst the body of knowledge is not perfect and is based on estimation, it is fundamental to the scientific method to assume the current body of knowledge we have is correct because it is all based on experimental evidence. 
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 10062013 00:41
(Original post by Mladenov)
Then again, by chinese reminder theorem, we can choose such that and . Thus the equation still has infinitely many solutions.
x=a(n)
x=b(m) but it is not generally ab.
In particular I could replace the 9 above with any other number in {1,2....2012} and you could still run the above CRT argument, but there are not always infinitely many solutions.
So obviously you mean something different to what I'm inferring?...
EDIT: Your edit makes it much clearer. Can you find an 'elementary' solution?Last edited by TheMagicMan; 10062013 at 00:45. 
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 10062013 00:44
(Original post by TheMagicMan)
How many 6th powers are there of form ? is more what I was aiming for.
Maybe as 6th powers. 
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 10062013 00:48
(Original post by TheMagicMan)
It is not entirely clear how this solves the problem. The CRT guarantees existence and uniqueness of a solution mod (mn) to a system
x=a(n)
x=b(m) but it is not generally ab.
In particular I could replace the 9 above with any other number in {1,2....2012} and you could still run the above CRT argument, but there are not always infinitely many solutions.
So obviously you mean something different to what I'm inferring?...
Set , . Then . It is sufficient to show that is solvable.
C'mon ; thus our problem has no solutions as are not cubic residues .Last edited by Mladenov; 10062013 at 00:58. 
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 10062013 00:54
(Original post by jack.hadamard)
So, you say Cambridge drilling (as William Hopkins' pupils described it) is quite good.
Maybe as 6th powers.Last edited by TheMagicMan; 10062013 at 00:57. 
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 10062013 00:55
(Original post by Mladenov)
Actually, what I am doing is:
Set , . Then . It is sufficient to show that is solvable. 
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 10062013 00:57
(Original post by Jkn)
But air resistance exists in the real world and the problem is taking place in that same world and not the mathematical world until we chose to create a model. Is it's current state, the problem exists only in the real world.
Proving that air resistance is nonnegligible is the objective of the first part of the problem (to allow us to consider the 'drops' as analogous to 'scales'). We prove it is nonnegligible by first assuming assuming all of the law of Physics and then approximating each law as is appropriate for the problem.
(Original post by Jkn)
We decide what is appropriate by the situation; if we wanted to we could use all of the compacted laws, it would just take longer and mean we must seek further information by going out and doing experiments or modelling the coins and the bag and certain objects whilst remarking the validity of this assumption. Is Physics, this is the way every problem must be done and you are certainly correct in saying that this method would require a document listing all assumptions along with experimental evidence as to their validity. When we do Physics, we do not write such documents because we call on experiments performs in similar situations to tell us things about what might happen in our current situations. We establish mathematical relationships between the variables recorded in an experiment and this is how we get our Physical laws. Note that all laws must be accompanied with an analysis of how well this correlates with experiment as well as when it is valid to apply these laws. These documents make up the body of Physics and are the very documents to which you refer.
Also I'm very much aware of at least how physical laws relate to experiments thank you very much, having taken the same Physics as NatScis in first year.Last edited by ukdragon37; 10062013 at 01:03. 
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 10062013 01:00

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 10062013 01:04

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 10062013 01:24
(Original post by ukdragon37)
Spoiler:ShowI've decided to give you a taste of what I do
Problem 224*/**:
Whenever we say "function" we mean total function.
Denote for every the set .
The mystery concept of a strange product for two such sets and satisfies the following property:
There exist functions and such that:
For each and each pair of functions and there exists a unique function such that for all we have and
Show that regardless of what the exact definition for is, if it satisfies the above property then for all sets and (nonzero) natural , iff (in other words ).
EDIT1: made it clearer, but question is still the same.
EDIT2: Hmmm I just realised that the elegant solution is far beyond *level, and the inelegant one may be quite long.
EDIT3: Actually for the above reason I'm going to withdraw it as a formal problem for this thread, but it's open to anyone who wants to try it.
It turns out that, if and are two different products, then . Let the projective maps be ; ; ; . Further, there are two uniquely determined mappings and .
Therefore,
It follows, from the uniqueness of , that ; similarly, .
Hence the result.
Spoiler:ShowMac Lane is sublime! 
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 10062013 01:30
(Original post by ukdragon37)
EDIT2: Hmmm I just realised that the elegant solution is far beyond *level, and the inelegant one may be quite long.
So, are you already at Cornell or are you about to go there?
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