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The 2012 STEP Results Discussion Thread

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  • View Poll Results: Should we include the AEA in this thread?
    Yes
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    No
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    66.24%

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    (Original post by DFranklin)

    Overall, I think this is great - I feel it has definitely levelled the playing field compared with when I did the exams, and you'd be competing against private/grammar schools that had specific classes for CCE preparation (and sl e of these schools would have libraries of past-papers/worked solutions going back decades).
    But if you go back another ten years (I took the entrance exam in '73) all state school applicants would have been at state grammar schools which would have the skills to prepare them for what was then a seventh-term exam. Compared with that system, I find the current way of doing things very discriminatory.
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    (Original post by dugdugdug)
    FYI, the guy who told me this is in his early 50s, if that's any help.

    BTW, how much harder is BMO compared to STEP and if someone is a member of the BMO, how much of a stretch is a degree? Why not go straight into a PhD, without a first degree, which I believe is legal.

    STEP<BMO<IMO so how can anyone "teach" an IMO candidate,
    BMO/IMO is a very different type of exam than STEP, so it's hard to compare. I think in all the BMO questions I've seen, I've only been able to do one or two completely unaided. (Then I looked up the AM-GM inequality and a few more yielded). I've only ever looked at one IMO paper and was dead proud to be able to do one question pretty easily (it's the one with the windmill, that question is not hard :confused:). I'm going to guess that is an anomaly. The aim is to stretch the brightest, not to teach them mathematics.

    The stuff you learn at university is very different - of course, many of those who do well at IMO/BMO do exceptionally well at university, but its definitely not a given. Everyone is different and they take to the more abstract way of thinking at university in different ways.

    Finally, it should be fairly obvious that the skills to excel at a PhD/research problem are different. Being able to produce a clever trick to solve a BMO problem is one thing, but being able to hack away at a problem which could take years to yield is quite another. How can someone with no training in mathematics straight from a very narrow syllabus cope with a PhD? They wouldn't even be able to understand the problem!

    I'm curious though, whether in an essay based subject, some young prodigy has gone straight to a PhD? I'm guessing not, for the same reasons as in maths, you still need to learn the theory and ideas of scholars past.

    (Original post by ian.slater)
    But if you go back another ten years (I took the entrance exam in '73) all state school applicants would have been at state grammar schools which would have the skills to prepare them for what was then a seventh-term exam. Compared with that system, I find the current way of doing things very discriminatory.
    What is 'the seventh term' and 'the fourth term'? I don't understand how its possible to have a 'pre-A Level' applicant

    I'm beginning to agree that going to a grammar school probably aided my education (as much as I hated that school). Thinking about the AEA, I wasn't really shocked or surprised by the sort of questions - in fact they seemed run-of-the-mill. And thinking about it, that was because we were given tests every week and the school must've slipped in old S-Level questions. They definitely wanted to push us (oddly I never felt very pushed, and hated the whole test a week thing) but it was a world apart from the comprehensive I went to for GCSEs.

    But the grammar school didn't really do much beyond give us the test. Solutions were never given (we would have to work it out for ourselves), and STEP class was apparently someone giving out a few years past papers and then saying go get going. But the fact there was a STEP class to begin with is much more than most people got in 2002... There definitely wasn't even any recognition for performance in the AEA. We discussed it in our first week in Year 13 and all I got was 'well it was a good thing we know how to recognise potential'.
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    (Original post by shamika)
    What is 'the seventh term' and 'the fourth term'? I don't understand how its possible to have a 'pre-A Level' applicant
    fourth term - first term of second year of A-level
    seventh term - taking the exam in the autumn after having completed A-levels the summer before
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    (Original post by dugdugdug)
    FYI, the guy who told me this is in his early 50s, if that's any help.

    BTW, how much harder is BMO compared to STEP and if someone is a member of the BMO, how much of a stretch is a degree? Why not go straight into a PhD, without a first degree, which I believe is legal.

    STEP<BMO<IMO so how can anyone "teach" an IMO candidate,
    You can't be a member of the BMO...and it's hardly preparation for a degree doing it
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    (Original post by shamika)
    I'm curious though, whether in an essay based subject, some young prodigy has gone straight to a PhD? I'm guessing not, for the same reasons as in maths, you still need to learn the theory and ideas of scholars past.

    I think to study for a phd you need a bachelor's in most countries (although I don't actually know if this is a legal requirement). I suppose you could be awarded an honorary phd without a bachelor's
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    (Original post by TheMagicMan)
    Spoiler:
    Show
    \displaystyle\frac{(\sqrt{ \cos(2 x)+\cos(4 x)}  \csc ( \frac{ \pi}{4}- \frac{x}{2}) \csc( \frac{ \pi}{4} + \frac{x}{2}) (2 \sin(x) \sqrt{2 \cos(2 x)-1}+\arctan( \frac{2 \sin(x)}{ \sqrt{2 \cos(2 x)-1}})}{8 \sqrt {2 \cos(2 x)-1}}
    Looks nice, doesn't it? I would offer a version for non-imaginary integrands.
    Spoiler:
    Show

    I don't think what follows is needed at all, but lets do it for fun.

    I = \displaystyle \int \sqrt{\cos(4x) + \cos(2x)}\ dx = \int \sqrt{2\cos^2(2x) + \cos(2x) - 1}\ dx

    \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \int \sqrt{(2\cos(2x) - 1)(\cos(2x) + 1)}\ dx

    \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{2} \int \sqrt{1 - 4\sin^2(x)}\ |\cos(x)|\ dx

    \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{2}\cdot \text{sgn}(\cos(x)) \int \sqrt{1 - 4\sin^2(x)}\ \cos(x)\ dx

    Hence, the integrand is imaginary when the quantity under the square root is negative, obviously.

    1 - 4\sin^2(x) &lt; 0 \Leftrightarrow \frac{1}{2} &lt; |\sin(x)|

    The above lives in the intervals \frac{\pi}{6} &lt; x &lt; \frac{5\pi}{6}, \frac{7\pi}{6} &lt; x &lt; \frac{11\pi}{6} for 0 \leq x \leq 2\pi.

    More generally, I'm interested in the intervals \frac{(3n - 1)\pi}{6} \leq x \leq \frac{(3n + 1)\pi}{6} for even n \in \mathbb{Z}, where the integrand is non-imaginary.


    Now, lets substitute 2\sin(x) = \sin(\omega), so that \cos(x)\ dx = \frac{1}{2}\cos(\omega)\ d\omega.

    \displaystyle \implies \int \sqrt{1 - 4\sin^2(x)}\ \cos(x)\ dx = \frac{1}{2} \int |\cos(\omega)|\ \cos(\omega)\ d\omega

    However, considering the intervals for x, it is easy to conclude that -\frac{\pi}{2} \leq \omega \leq \frac{\pi}{2}, so that \cos(\omega) \geq 0

    \displaystyle \therefore \int \sqrt{1 - 4\sin^2(x)}\ \cos(x)\ dx = \frac{1}{2} \int \cos^2(\omega)\ d\omega = \frac{1}{4}(\omega + \sin(\omega)\cos(\omega)) + c

    Finally, to bring everything back.

    \displaystyle \therefore \ I = \frac{\text{sgn}(\cos(x))}{2 \sqrt{2}}\left[\sin^{-1}(2\sin(x)) + 2\sin(x)\sqrt{1 - 4\sin^2(x)} \right] + c

    I can't believe I actually typed this in. Lets have some fun.

    \displaystyle \int_{n - \frac{\pi}{6}}^{n + \frac{\pi}{6}} \sqrt{\cos(4x) + \cos(2x)}\ dx = \frac{\pi}{2\sqrt{2}} for every n \in \mathbb{Z}.

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    (Original post by TheMagicMan)
    You can't be a member of the BMO...and it's hardly preparation for a degree doing it
    I think he meant British IMO team
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    (Original post by shamika)
    I'm curious though, whether in an essay based subject, some young prodigy has gone straight to a PhD? I'm guessing not, for the same reasons as in maths, you still need to learn the theory and ideas of scholars past.
    There are people getting, or starting, their PhD's at the age of 18...
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    I still need to prepare for STEP how hard is it to achieve a 2?
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    (Original post by gymrat)
    I still need to prepare for STEP how hard is it to achieve a 2?
    I'm with the impression that if you're managing three solutions and a fourth partially, you're guaranteed a 2, if not more.
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    (Original post by shamika)
    BMO/IMO is a very different type of exam than STEP, so it's hard to compare. I think in all the BMO questions I've seen, I've only been able to do one or two completely unaided. (Then I looked up the AM-GM inequality and a few more yielded). I've only ever looked at one IMO paper and was dead proud to be able to do one question pretty easily (it's the one with the windmill, that question is not hard :confused:). I'm going to guess that is an anomaly. The aim is to stretch the brightest, not to teach them mathematics.

    The stuff you learn at university is very different - of course, many of those who do well at IMO/BMO do exceptionally well at university, but its definitely not a given. Everyone is different and they take to the more abstract way of thinking at university in different ways.

    Finally, it should be fairly obvious that the skills to excel at a PhD/research problem are different. Being able to produce a clever trick to solve a BMO problem is one thing, but being able to hack away at a problem which could take years to yield is quite another. How can someone with no training in mathematics straight from a very narrow syllabus cope with a PhD? They wouldn't even be able to understand the problem!

    I'm curious though, whether in an essay based subject, some young prodigy has gone straight to a PhD? I'm guessing not, for the same reasons as in maths, you still need to learn the theory and ideas of scholars past.



    What is 'the seventh term' and 'the fourth term'? I don't understand how its possible to have a 'pre-A Level' applicant

    I'm beginning to agree that going to a grammar school probably aided my education (as much as I hated that school). Thinking about the AEA, I wasn't really shocked or surprised by the sort of questions - in fact they seemed run-of-the-mill. And thinking about it, that was because we were given tests every week and the school must've slipped in old S-Level questions. They definitely wanted to push us (oddly I never felt very pushed, and hated the whole test a week thing) but it was a world apart from the comprehensive I went to for GCSEs.

    But the grammar school didn't really do much beyond give us the test. Solutions were never given (we would have to work it out for ourselves), and STEP class was apparently someone giving out a few years past papers and then saying go get going. But the fact there was a STEP class to begin with is much more than most people got in 2002... There definitely wasn't even any recognition for performance in the AEA. We discussed it in our first week in Year 13 and all I got was 'well it was a good thing we know how to recognise potential'.
    I say a PhD without a bachelors because I recall watching a programme on Channel 4 about prodigies and one was doing just that (though his mum was a professor) and it was in Chemistry.

    In the old days, Oxford had an entrance exam in the fourth term and Cambridge in the seventh term, ie as someone remarked, after completing A-Levels, one had to return for the Cambridge exam and pass or fail, one would have to take a year out.

    In the REALLY old days, someone said Cambridge also had a fourth term paper.
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    (Original post by hassi94)
    I think he meant British IMO team
    Excuse my ignorance, (I was only in the Hans Woyda London Maths competition!) but if one is in the BMO team, that means they're representing the UK in the IMO?
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    (Original post by ian.slater)
    But if you go back another ten years (I took the entrance exam in '73) all state school applicants would have been at state grammar schools which would have the skills to prepare them for what was then a seventh-term exam. Compared with that system, I find the current way of doing things very discriminatory.
    Ah, a senior member of these columns! Do you recall a book called Pure Maths, vols 1 and 2 by S L Parsonson of Harrow School? Those questions were deadly (for me anyway).
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    (Original post by dugdugdug)
    Excuse my ignorance, (I was only in the Hans Woyda London Maths competition!) but if one is in the BMO team, that means they're representing the UK in the IMO?
    The BMO is just an exam really. So there's no such thing as BMO team (I don't think - it's specifically the British team for the IMO) and you certainly can't be 'a member of the BMO' because that's like being 'a member of the STEP'.
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    (Original post by hassi94)
    The BMO is just an exam really. So there's no such thing as BMO team (I don't think - it's specifically the British team for the IMO) and you certainly can't be 'a member of the BMO' because that's like being 'a member of the STEP'.
    So how do you do the BMO, (you just submit an application like STEP?) and how is Britain represented in the IMO? Who gets to choose who gets into the team or is it based on the results of the BMO?
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    (Original post by dugdugdug)
    So how do you do the BMO, (you just submit an application like STEP?) and how is Britain represented in the IMO? Who gets to choose who gets into the team or is it based on the results of the BMO?
    I think it gets cut further and further. So

    UKMT < BMO1 < BMO2 then from those some get picked to go to Cambridge or something where they're cut even further and then a final team is assembled.
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    (Original post by dugdugdug)
    So how do you do the BMO, (you just submit an application like STEP?) and how is Britain represented in the IMO? Who gets to choose who gets into the team or is it based on the results of the BMO?
    Well you automatically get into the BMO1 free if you do well enough on Senior UKMT. Otherwise, you can pay a small fee to enter. Then the top people from BMO1 go to BMO2. Then top people on BMO2 get selected to try out for the IMO team I think.
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    (Original post by shamika)
    I've only ever looked at one IMO paper and was dead proud to be able to do one question pretty easily (it's the one with the windmill, that question is not hard :confused:).
    http://www.artofproblemsolving.com/F...759ba8767782d7
    Q2?

    I can't even understand it. You should try some more, you might have a talent. :rolleyes:
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    By all means IMO questions are interesting, and by no means going to the IMO should be your top priority, necessarily.

    I know a guy who few years ago won a Gold medal, absolute ranking 1 (with 100%), went to Harvard for Maths with Economics, and left after bachelors to become Economist.
    This olympiad does not show that you have a passion for Maths, or even that you'll be a great mathematician -- all it shows is potential and how good olympiad training is.

    However, it is the best marketplace for young mathematicians. If you want to sell yourself, that's the place. Simple.

    Spoiler:
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    I think this is useful, because I have seen so many people actually believing that this thing is the end of the world.
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    (Original post by gff)
    By all means IMO questions are interesting, and by no means going to the IMO should be your top priority, necessarily.

    I know a guy who few years ago won a Gold medal, absolute ranking 1 (with 100%), went to Harvard for Maths with Economics, and left after bachelors to become Economist.
    This olympiad does not show that you have a passion for Maths, or even that you'll be a great mathematician -- all it shows is potential and how good olympiad training is.

    However, it is the best marketplace for young mathematicians. If you want to sell yourself, that's the place. Simple.

    Spoiler:
    Show

    I think this is useful, because I have seen so many people actually believing that this thing is the end of the world.
    How the hell you know this guy?

    And how IMO people can sell themselves?

    I heard something like this about coding competitions (or olympiads, don't know how they're called), but it's fairly obvious who can be interested in them.

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Updated: November 18, 2012
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