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# Girls vs boys maths challenge watch

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1. (Original post by Smaug123)
Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something .
It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.
2. (Original post by Smaug123)
Or the arguably non-existent r=0
Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something - it sounds vaguely relevant and a little bit harder level
You might or might not find it helpful to know Heron's formula for the area of a triangle: that if , then .
Christ, feeling puny with my piddly little GCSE and Add Maths

But I'll give it a shot...

Any hints on where to start?

3. (Original post by Jkn)
Ohh interesting. He researches gen. rel., right? Oh that's awesome. If I've somehow scraped my way in I may well end up drifting towards that department.
Nooooo, DAMTP is the dark side! Pure is the way forward Yep, he's GR. "Scraped in" is perhaps a bit pessimistic, from the look of it :P

Interrogation: What year are you in at the moment then? How are you finding the course and/or coping with it? What areas most interest you? Next year options? etc.. etc.. etc..
Edit: Btw, on the above example, what does 'variational principles' offer? Is this some crazy generalisation using imaginary/non-euclidean metric spaces? Can one not use the three techniques that spring to mind? (leaving it for the younger ones though, of course, unless you make us look up variational principles to try and 'wing' a proof).
I'm going into the second year, loving the course I didn't do enough work earlier in the year, so it was a big rush before the exams, but I've got a better work ethic/schedule now. Group theory is a great thing, I love GRM (Groups, Rings and Modules, it's in Part IB but loads of first years take the lectures early). I'm definitely a puremo! [in case you didn't know, mathematicians are "mathmos", and the term is extensible - "appliedmo", "puremo", "Trinmo" for Trinity, etc] I'm avoiding everything applied, apart from quantum, which is as close to compulsory as they come.
Variational principles is just a more complicated way to get the answer - it would be a disgusting method in this case. It's simply a way of extremising for a,b,f fixed. That is, you use it to find y, z, etc such that I[x] is extremised. It's a yucky but very general way to do this kind of thing
Nah, we've not really done anything explicitly in non-euclidean metric spaces - the closest we've come is topological spaces, which we've not covered in much depth (that's a IB course too). Almost everything applied has been in for some n…
Three techniques that spring to mind - I can only think of one plausible one (divide the thing up into two right-angled triangles), but my problem-solving method is: "Generate a possible line of attack. See if line of attack works. Repeat." so I would usually only have one line of attack at a given time, even on such a simple problem. Actually, the answer is clearly at least isosceles by the most rudimentary application of symmetry, so there's several more lines of attack opened up that would simplify a variational calculus proof a lot, too, knowing that a=b.
Bleugh, late at night and incoherent…
4. (Original post by BabyMaths)
It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.
Phew

Christ, feeling puny with my piddly little GCSE and Add Maths
But I'll give it a shot...
Any hints on where to start?
Apologies, I posed it badly - I meant to say "the triangle with fixed perimeter that maximises the area is equilateral".
Actually, this is a fairly nice way to extoll the virtues of symmetry
OK, so I've told you that the area of the triangle is that expression. The area is fixed, so let's call it A. (It's the same A as in that expression.) Let's call the side lengths a,b,c.
Now, what are you trying to show? Can you give me a couple of equations that sum up the problem "Prove that the triangle with area A should be equilateral"? (I realise it's quite hard to tell what I mean by that - say if you don't understand!)
5. (Original post by Smaug123)
Pretty much every applied example sheet we get, he says "Oh, I wrote that question five years ago" or similar on at least two questions :P I'm not sure if there actually is a maths department at all - it's just Siklos and about fifty figureheads…
I wish I applied to Cambridge instead, I think I would've stood a better chance than Oxford...back when I did the MAT I didn't realise that I should've had extra time...most of the questions I did were right, I just didn't do enough so I didn't get an interview

do you know much about applying to cambridge for postgrad? I was thinking of doing that for my Masters or something like that but I don't know how it works :/
6. (Original post by LeeMrLee)
do you know much about applying to cambridge for postgrad? I was thinking of doing that for my Masters or something like that but I don't know how it works :/
Have you completed/are you shortly to complete an undergrad course? If not, don't worry about this yet - know only that it is pretty common to take the Masters at Cambridge without having studied at Cambridge before
If you are in a position where it would be useful to know, I'm afraid my knowledge of Part III admissions is somewhat limited. http://www.maths.cam.ac.uk/postgrad/mathiii/ might be helpful.
7. (Original post by Smaug123)
Have you completed/are you shortly to complete an undergrad course? If not, don't worry about this yet - know only that it is pretty common to take the Masters at Cambridge without having studied at Cambridge before
If you are in a position where it would be useful to know, I'm afraid my knowledge of Part III admissions is somewhat limited. http://www.maths.cam.ac.uk/postgrad/mathiii/ might be helpful.
No, I'm about to start my first degree in September xD I'm only asking because I'm planning ahead different possible routes I can take. Thanks for the link

It says I need to get a first as one of the requirements, so I suppose I should see how my undergrad degree goes first and if it's going well then start planning for postgrad :L
8. (Original post by LeeMrLee)
No, I'm about to start my first degree in September xD I'm only asking because I'm planning ahead different possible routes I can take. Thanks for the link

It says I need to get a first as one of the requirements, so I suppose I should see how my undergrad degree goes first and if it's going well then start planning for postgrad :L
Yeah, and the masters is also aimed solely at people who want to go on for research - not just for an extra qualification.
9. (Original post by Smaug123)
Yeah, and the masters is also aimed solely at people who want to go on for research - not just for an extra qualification.
at the moment, I do want to go into some sort of research role, either that or teaching. I enjoy teaching maths stuff to people, I spend a lot of time with friends going over things they don't understand in A-level so I want to maybe go into teaching maybe at A-level FM or harder sort of level.
E, C and A?
11. (Original post by Smaug123)
Nooooo, DAMTP is the dark side! Pure is the way forward Yep, he's GR. "Scraped in" is perhaps a bit pessimistic, from the look of it :P
I'm very unsure. I love mathematics as an objective art (or because because it is 'the' objective art) but have lately realised that lots of Calculus/Analysis 'apparently' is labelled as 'applied'?! I also love Physics though have never been too keen on Newtonian Mechanics (I feel as though, if I'm going to do ugly maths, then it better damn well mean something!) I do, of course, have no idea which I will opt for at the moment.. hopefully the next year or two will be long enough to decide.
I'm going into the second year, loving the course I didn't do enough work earlier in the year, so it was a big rush before the exams, but I've got a better work ethic/schedule now.
Awesome. Why not enough work earlier in the year? Social life, etc.. ? Btw, do you ever get to just hang about a chalkboard (with other students) doing random maths? And also, do you meet loads of brilliant mathematicians?
Group theory is a great thing, I love GRM (Groups, Rings and Modules, it's in Part IB but loads of first years take the lectures early). I'm definitely a puremo! [in case you didn't know, mathematicians are "mathmos", and the term is extensible - "appliedmo", "puremo", "Trinmo" for Trinity, etc] I'm avoiding everything applied, apart from quantum, which is as close to compulsory as they come.
I am yet to be impressed by group theory. It seems that most of the significant applications and links to other areas of maths that can be drawn from it are rather inaccessible (none the less, I persist..). Would love it if you have any examples of it's power and/or importance?

"Trinmo" lmao, please say that's not seen as some sort of 'status'?

QM all the way!
Variational principles is just a more complicated way to get the answer - it would be a disgusting method in this case. It's simply a way of extremising for a,b,f fixed. That is, you use it to find y, z, etc such that I[x] is extremised. It's a yucky but very general way to do this kind of thing
Nah, we've not really done anything explicitly in non-euclidean metric spaces - the closest we've come is topological spaces, which we've not covered in much depth (that's a IB course too). Almost everything applied has been in for some n…
Oh god, I need to do so much vectors over the summer..
Three techniques that spring to mind - I can only think of one plausible one (divide the thing up into two right-angled triangles), but my problem-solving method is: "Generate a possible line of attack. See if line of attack works. Repeat." so I would usually only have one line of attack at a given time, even on such a simple problem. Actually, the answer is clearly at least isosceles by the most rudimentary application of symmetry, so there's several more lines of attack opened up that would simplify a variational calculus proof a lot, too, knowing that a=b.
Bleugh, late at night and incoherent…
(Original post by BabyMaths)
It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.
Well the "three techniques" I had in mind were: using calculus to maximise area (using the sine definition or Heron's) which could get messy, AM-GM on Heron's or Lagrange Multipliers.

Actually, come to think of it, perhaps M3/M5-dodgy-limit-calculus could be used? i.e. maximising M.O.I and similar things and making deductions etc.. hmm..

Integration should work too..

Hold on.. you say 'maximise for a given area'. I don't think that makes sense? Is this subject to the sum of the lengths being constant or something like that?
12. (Original post by Smaug123)
Nooooo, DAMTP is the dark side! Pure is the way forward Yep, he's GR. "Scraped in" is perhaps a bit pessimistic, from the look of it :P
I'm very unsure. I love mathematics as an objective art (or because because it is 'the' objective art) but have lately realised that lots of Calculus/Analysis 'apparently' is labelled as 'applied'?! I also love Physics though have never been too keen on Newtonian Mechanics (I feel as though, if I'm going to do ugly maths, then it better damn well mean something!) I do, of course, have no idea which I will opt for at the moment.. hopefully the next year or two will be long enough to decide.

Well I might have cocked up STEP.. and unfortunately there isn't much else..
I'm going into the second year, loving the course I didn't do enough work earlier in the year, so it was a big rush before the exams, but I've got a better work ethic/schedule now.
Awesome. Why not enough work earlier in the year? Social life, etc.. ? Btw, do you ever get to just hang about a chalkboard (with other students) doing random maths? And also, do you meet loads of brilliant mathematicians?
Group theory is a great thing, I love GRM (Groups, Rings and Modules, it's in Part IB but loads of first years take the lectures early). I'm definitely a puremo! [in case you didn't know, mathematicians are "mathmos", and the term is extensible - "appliedmo", "puremo", "Trinmo" for Trinity, etc] I'm avoiding everything applied, apart from quantum, which is as close to compulsory as they come.
I am yet to be impressed by group theory. It seems that most of the significant applications and links to other areas of maths that can be drawn from it are rather inaccessible (none the less, I persist..). Would love it if you have any examples of it's power and/or importance?

"Trinmo" lmao, please say that's not seen as some sort of 'status'?

QM all the way!
Variational principles is just a more complicated way to get the answer - it would be a disgusting method in this case. It's simply a way of extremising for a,b,f fixed. That is, you use it to find y, z, etc such that I[x] is extremised. It's a yucky but very general way to do this kind of thing
Nah, we've not really done anything explicitly in non-euclidean metric spaces - the closest we've come is topological spaces, which we've not covered in much depth (that's a IB course too). Almost everything applied has been in for some n…
Oh god, I need to do so much vectors over the summer..
Three techniques that spring to mind - I can only think of one plausible one (divide the thing up into two right-angled triangles), but my problem-solving method is: "Generate a possible line of attack. See if line of attack works. Repeat." so I would usually only have one line of attack at a given time, even on such a simple problem. Actually, the answer is clearly at least isosceles by the most rudimentary application of symmetry, so there's several more lines of attack opened up that would simplify a variational calculus proof a lot, too, knowing that a=b.
Bleugh, late at night and incoherent…
(Original post by BabyMaths)
It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.
Well the "three techniques" I had in mind were: using calculus to maximise area (using the sine definition or Heron's) which could get messy, AM-GM on Heron's or Lagrange Multipliers.

Actually, come to think of it, perhaps M3/M5-dodgy-limit-calculus could be used? i.e. maximising M.O.I and similar things and making deductions etc.. hmm..

Integration should work too..

Hold on.. you say 'maximise for a given area'. I don't think that makes sense? Is this subject to the sum of the lengths being constant or something like that?
13. (Original post by Smaug123)
Phew

Apologies, I posed it badly - I meant to say "the triangle with fixed perimeter that maximises the area is equilateral".
Actually, this is a fairly nice way to extoll the virtues of symmetry
OK, so I've told you that the area of the triangle is that expression. The area is fixed, so let's call it A. (It's the same A as in that expression.) Let's call the side lengths a,b,c.
Now, what are you trying to show? Can you give me a couple of equations that sum up the problem "Prove that the triangle with area A should be equilateral"? (I realise it's quite hard to tell what I mean by that - say if you don't understand!)
I thought about dividing it into two right-angled triangles and going from there, but I ended up with a pile of gibberish. Attempted calculus, but I ended up with a pile of gibberish. AS level is going to go reeeeeally well at this rate.
14. (Original post by Jkn)
I'm very unsure. I love mathematics as an objective art (or because because it is 'the' objective art) but have lately realised that lots of Calculus/Analysis 'apparently' is labelled as 'applied'?! I also love Physics though have never been too keen on Newtonian Mechanics (I feel as though, if I'm going to do ugly maths, then it better damn well mean something!) I do, of course, have no idea which I will opt for at the moment.. hopefully the next year or two will be long enough to decide.
Most of the calculus is fairly applied - Analysis is pure, although it can stretch to "applied" depending on how much you use it to make methods like the chain rule. Unfortunately, all of the dynamics in Dynamics+Relativity is Newtonian we do a whistlestop tour of Lagrangian in Variational Principles. Yeah, the next year will tell you whether you're a puremo or appliedmo

Well I might have cocked up STEP.. and unfortunately there isn't much else..
I thought that, and I got either an S or a 1 in STEP III (I thought I was scraping a 2 - I spent ages on Q1 which had the first line missing) - and there's always the pool, if nothing else

Awesome. Why not enough work earlier in the year? Social life, etc.. ? Btw, do you ever get to just hang about a chalkboard (with other students) doing random maths? And also, do you meet loads of brilliant mathematicians?
A bit, mainly just the fact that I'd never really done something that I'd had to work at before (STEP doesn't count, as that wasn't taught) - I didn't have the habits set up to work well. Lots of hanging around a whiteboard (a great investment was http://www.magicwhiteboardproducts.c...25-sheet-roll/ if you can find it - Ryman's stocks them, but quite expensively - even the Natscis used them all the time!) doing lots of random maths. There are some brilliant mathematicians - it's really quite worrying how good some of them are :P

I am yet to be impressed by group theory. It seems that most of the significant applications and links to other areas of maths that can be drawn from it are rather inaccessible (none the less, I persist..). Would love it if you have any examples of it's power and/or importance?
Oh, I don't know of any significant applications of group theory at the moment, apart from a vague knowledge that it is used in crystallography and that ring theory (closely related) can be used to show that precisely {the odd primes 1 mod 4, and the numbers in whose prime factorisation any primes 3 mod 4 occur to only even powers} can be expressed as a sum of two squares. (We did that in lectures.) It's just really really pretty!

"Trinmo" lmao, please say that's not seen as some sort of 'status'?
Nah, it's mildly derogatory

Oh god, I need to do so much vectors over the summer..
It's all taught from scratch - you'll be fine with essentially no work over the summer (I didn't do anything!)

Actually, come to think of it, perhaps M3/M5-dodgy-limit-calculus could be used? i.e. maximising M.O.I and similar things and making deductions etc.. hmm..
Yeah, I've just done it with Lagrange multipliers as an exercise on a Variational Principles example sheet. I thought I had a nice symmetry argument, but it's actually rubbish.

Hold on.. you say 'maximise for a given area'. I don't think that makes sense? Is this subject to the sum of the lengths being constant or something like that?
Yeah, I amended a later post to read "maximise a given area for constant perimeter"
I thought about dividing it into two right-angled triangles and going from there, but I ended up with a pile of gibberish. Attempted calculus, but I ended up with a pile of gibberish. AS level is going to go reeeeeally well at this rate.
Don't worry - it turns out that the problem is quite hard using GCSE maths. AS and A-level is nothing like this - they'll as good as tell you exactly what to do in each question
I'll give you a step-by-step answer instead - do you know about partial differentiation? (Or any differentiation at all - we can work with that.)
16. (Original post by Smaug123)
Don't worry - it turns out that the problem is quite hard using GCSE maths. AS and A-level is nothing like this - they'll as good as tell you exactly what to do in each question
I'll give you a step-by-step answer instead - do you know about partial differentiation? (Or any differentiation at all - we can work with that.)
Partial differentiation at GCSE o.O that seems rather severe
17. (Original post by Smaug123)
Don't worry - it turns out that the problem is quite hard using GCSE maths. AS and A-level is nothing like this - they'll as good as tell you exactly what to do in each question
I'll give you a step-by-step answer instead - do you know about partial differentiation? (Or any differentiation at all - we can work with that.)
Hey! Could you possibly explain that partial differentiation please? I've done C1,C2 and have just started C3. I can differentiate e^x, ln x, sin, tan, cos and using the chain, product and quotient rules if that helps. Thanks
18. (Original post by Smaug123)
Or the arguably non-existent r=0
Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something - it sounds vaguely relevant and a little bit harder level
You might or might not find it helpful to know Heron's formula for the area of a triangle: that if , then .
I think I'll go down the Lagrange multipliers route as I suck at geometry
Spoiler:
Show

Note that is maximised is also maximised (or minimised if negative and )
Assuming A is positive for the triangle, then this only occurs when A is maximised.
The constraint on the triangle is that where P is some positive constant.
Let:

Setting:

and considering:
we find that:

Applying a similar argument to:

Is the condition necessary for the only stationary point.
To demonstrate that this is a maximum (other than by common sense as there has to be a maximum for an area of a triangle with fixed perimeter.) is currently beyond me
19. (Original post by PhysicsKid)
Hey! Could you possibly explain that partial differentiation please? I've done C1,C2 and have just started C3. I can differentiate e^x, ln x, sin, tan, cos and using the chain, product and quotient rules if that helps. Thanks
Great, that's all we need
OK, so the key point is that the differentiation you've learnt is like:
.
We want a nice consistent way to define differentiation in the case of (for example) .
There's a very simple way we do this:
.
That is, we differentiate as if everything is constant except the variable we're differentiating with respect to. I'll now explain this better :P
Now, the key thing is that, just as a stationary point of can be found by differentiating and setting to 0 - that is, so is the stationary point - so we can find a stationary point of by partial-differentiating and setting to 0. Think of it as: we're making a 3D diagram with x and y on the x and y axes, and with height representing f(x,y) at the point (x,y). Then any stationary point (for example, a minimum - that looks like a bowl on our 3D diagram) must be stationary if we slice through the x-axis or the y-axis.
Example series of pictures:
Spoiler:
Show
Let's consider the function .

Now, let's take a slice through the x-axis (so we're looking at it side-on):

Here it is with the right-hand half of the slice removed:

And here is what it looks like from the side on (it extends backwards into the page, but the pertinent bit is this bit):

Notice that the stationary point in the 3D diagram must be stationary if we take a 2D slice through it, as it is in this slice.

Now, the key thing is that we can easily do this "slicing" just by setting x or y to be constant (in the example, I've set x=0). But if x is constant, then if we differentiate with respect to y, we don't care about x - it just behaves as if it were (say) 2.
So , because the x^2 was constant so when we differentiated, it became 0. , because y was constant so sin(y) was constant.

OK so far? That's partial differentiation in a nutshell - my next post, if you're with me so far, will be how to solve the triangles question using this.
20. (Original post by Smaug123)
Great, that's all we need
OK, so the key point is that the differentiation you've learnt is like:
.
We want a nice consistent way to define differentiation in the case of (for example) .
There's a very simple way we do this:
.
That is, we differentiate as if everything is constant except the variable we're differentiating with respect to. I'll now explain this better :P
Now, the key thing is that, just as a stationary point of can be found by differentiating and setting to 0 - that is, so is the stationary point - so we can find a stationary point of by partial-differentiating and setting to 0. Think of it as: we're making a 3D diagram with x and y on the x and y axes, and with height representing f(x,y) at the point (x,y). Then any stationary point (for example, a minimum - that looks like a bowl on our 3D diagram) must be stationary if we slice through the x-axis or the y-axis.
Example series of pictures:
Spoiler:
Show
Let's consider the function .

Now, let's take a slice through the x-axis (so we're looking at it side-on):

Here it is with the right-hand half of the slice removed:

And here is what it looks like from the side on (it extends backwards into the page, but the pertinent bit is this bit):

Notice that the stationary point in the 3D diagram must be stationary if we take a 2D slice through it, as it is in this slice.

Now, the key thing is that we can easily do this "slicing" just by setting x or y to be constant (in the example, I've set x=0). But if x is constant, then if we differentiate with respect to y, we don't care about x - it just behaves as if it were (say) 2.
So , because the x^2 was constant so when we differentiated, it became 0. , because y was constant so sin(y) was constant.

OK so far? That's partial differentiation in a nutshell - my next post, if you're with me so far, will be how to solve the triangles question using this.
So when doing d/dy, x is constant and when doing d/dx, y is constant. When you have two terms added, the constant becomes 0 when differentiating since the constant has the same effect no matter what the other variable is. When terms are multiplied, the constant variable remains the same because if f(x) =3x^2 then dy/dx = 6x- the same as if you say take dy/dx of the variable x^2 (2x), and factor in the original multiplier effect of 3?

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