Girls vs boys maths challenge

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There's not. Before I thought of directly manipulating the Gamma function, I derived several infinite series for phi.

Here are a few possibilities of the kind that arise trivially:

\displaystyle \phi = 2 \cos \frac{\pi}{5} = \sum_{n=0}^{\infty} \frac{2 (-1)^n \pi^{2n}}{5^{2n} (2n)!}

\displaystyle \phi = \frac{1}{2} \sec \frac{2 \pi}{5} = \sum_{n=0}^{\infty} \frac{2^{2n-1} (-1)^n E_{2n} \pi^{2n}}{5^{2n} (2n)!}

Also, one more famous one is:

\displaystyle \phi = \frac{13}{8} + \sum_{n=0}^{\infty} \frac{(-1)^{n+1} (2n+1)!}{(n+1)! n! 4^{2n+3}}

A more beautiful one:

\displaystyle \phi = 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{F_n F_{n+1}}

etc..

I guess I was misinformed then. Mine is still another new addition though.

Reply 141

Jkn

Original post by peter12345

I guess I was misinformed then. Mine is still another new addition though.

How do you expect me to believe that you made it if you haven't shown a derivation/proof? :tongue:

Reply 142

peter12345

Original post by Jkn

How do you expect me to believe that you made it if you haven't shown a derivation/proof? :tongue:

Send me your email and I'll email it to you. It's like 5 pages long.

Reply 143

Jkn

Original post by 0x2a

x = \frac{\pi}{4}

Original post by PhysicsKid

x = 45

I think the question was flawed in that it asked for "the" value. In fact, as I'm sure you know, there are many more values of x for which

\sin x = \cos x

Reply 144

Jkn

Original post by peter12345

Send me your email and I'll email it to you. It's like 5 pages long.

I will give you it via. a PM. :tongue:

Reply 145

PhysicsKid

Original post by Jkn

I think the question was flawed in that it asked for "the" value. In fact, as I'm sure you know, there are many more values of x for which

\sin x = \cos x

Of course

But my substitution method only yields 45, which is the method given I said GCSE I expected to be used :wink:

Reply 146

0x2a

The part about this thread being girls vs boys is completely redundant now isn't it?

Problem

Find all two-digit integers N for which the sum of the digits of

10^N - N

is divisible by 170.

(edited 10 years ago)

Reply 147

LeeMrLee

Original post by 0x2a

The part about this thread being girls vs boys is completely redundant now isn't it?

judging from the beginning posts which were full of more basic questions (AS-level) I think whoever created this thread was possibly expecting A level difficulty questions xD I like the harder questions though even though I've not been able to do many (phone can't read latex and I'm lazy to do it myself). its been good to have a think about it though

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Reply 148

Felix Felicis

Original post by Jkn

Problem: Prove that

\displaystyle \phi = \frac{1}{\sqrt[4]{5}} \frac{\Gamma \left( \frac{1}{20} \right) \Gamma \left( \frac{9}{20} \right)}{\Gamma \left( \frac{3}{20} \right)\Gamma \left( \frac{7}{20} \right)}

I finally get to use the reflection formula :ninja:

\displaystyle \begin{aligned} \frac{1}{5^{\frac{1}{4}}} \cdot \frac{\Gamma \left( \frac{1}{20} \right) \Gamma \left( \frac{9}{20} \right)}{\Gamma \left( \frac{3}{20} \right) \Gamma \left( \frac{7}{20} \right)} & = \frac{1}{5^{\frac{1}{4}}} \cdot \frac{\Gamma \left( \frac{1}{20} \right) \Gamma \left( \frac{9}{20} \right) \Gamma \left( \frac{11}{20} \right) \Gamma \left( \frac{15}{20} \right) \Gamma \left( \frac{19}{20} \right)}{\Gamma \left( \frac{3}{20} \right) \Gamma \left( \frac{7}{20} \right) \Gamma \left( \frac{11}{20} \right) \Gamma \left( \frac{15}{20} \right) \Gamma \left( \frac{19}{20} \right)} \\ & = \frac{1}{5^{\frac{1}{4}}} \cdot \frac{\Gamma \left( \frac{3}{4} \right) \pi^{2} \csc \left( \frac{\pi}{20} \right) \csc \left( \frac{9 \pi}{20} \right)}{4 \pi^{2} 5^{ - \frac{1}{4}} \Gamma \left( \frac{3}{4} \right)} \\ & = \frac{\csc \left( \frac{\pi}{20} \right) \csc \left( \frac{9 \pi}{20} \right)}{4} \\ & = \frac{1}{4} \csc \left( \frac{\pi}{20} \right) \sec \left( \frac{\pi}{20} \right) \\ & = \frac{1}{2} \csc \left( \frac{\pi}{10} \right) \\ & = \frac{1 + \sqrt{5}}{2} \end{aligned}

as required.

Last part was done using de Moivre's but I can't be bothered typing it out :lol:

(though I will if anyone wants).

Edit: transition from first to second line:

The numerator was simplified by applying Euler's reflection formula twice on

\Gamma \left( \frac{1}{20} \right) , \ \Gamma \left( \frac{19}{20} \right) , \ \Gamma \left( \frac{9}{20} \right) , \ \Gamma \left( \frac{11}{20} \right)

The denominator was simplified by applying the multiplication theorem of

\prod_{r=1}^{4} \Gamma \left( \frac{3}{20} + \frac{r}{5} \right)

(edited 10 years ago)

Reply 149

Ateo

Original post by 0x2a

The part about this thread being girls vs boys is completely redundant now isn't it?

Problem

Find all two-digit integers N for which the sum of the digits of

10^N - N

is divisible by 170.

I am not satisfied with this solution for it lacks rigor and is essentially trial and error.

The sum of digits will be 9(N-2) + digitsum(100 - N)

Let

N=10a_1 + a_0 , 1\leq a_0,a_1\leq9

. We want;

9(10a_1 + a_0 - 2) + 19 - a_1 - a_0 = 89a_1 + 8a_0 + 1 = 0

(mod 170)

We then check for values of

a_0

from 1 through to 9 giving N = 39, 58, 77, 96
(It is fairly easy to find

a_1

since we are looking for the value that will make the digit sum end with a 0)

We must also check the case when

a_0 = 0

9(10a_1 - 2) + 10 - a_1 = 0

(mod 170) giving N = 20

I would be interested in a more elegant solution. Somebody posted that N has the form 1+19k for k = 1,2,3,4,5 but I can't see how to show this from the initial question. That post is gone now.

I think it would be better to post some of these questions in 'The proof is trivial' as * problems and let this thread go.

(edited 10 years ago)

Reply 150

Jkn

Original post by Felix Felicis

You're going to need more detail to show you you got from your first line to your second. :tongue:

Reply 151

0x2a

Original post by Ateo

I think it would be better to post some of these questions in 'The proof is trivial' as * problems and let this thread go.

Well I'm scared to post in that thread since all I see are integrals that go right over my head. :tongue:

Reply 152

LeeMrLee

Original post by 0x2a

Well I'm scared to post in that thread since all I see are integrals that go right over my head. :tongue:

same I'm not good enough for that one yet xD there should be an easier proof thread for noobs like us :wink:

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Reply 153

Jkn

Original post by 0x2a

Well I'm scared to post in that thread since all I see are integrals that go right over my head. :tongue:

Original post by LeeMrLee

same I'm not good enough for that one yet xD there should be an easier proof thread for noobs like us :wink:

Sorry guys, that's probably my bad. I set a good chunk of the problems on that thread (people are slacking :mad:

) and I've been getting really into integrals lately. :colone:

I posted a new set of problems up a few minutes ago if you guys are interested (however all but 2 or 3 are extremely difficult). If you have any requests for the kind of problems you would like to see, feel free to ask me on here, on there or by PM (I have a stupid amount of resources!) I don't want people to feel excluded! (the only problem is that easier problems tend to get solved instantly and so are ultimately discouraged). :tongue:

Reply 154

0x2a

Original post by Jkn

Sorry guys, that's probably my bad. I set a good chunk of the problems on that thread (people are slacking :mad:

) and I've been getting really into integrals lately. :colone:

Nope, still can't do them. :colondollar:

Don't worry though, I only learnt integration by parts like two months ago and I have eons till I need to prepare for STEP or anything close to it. :biggrin:

Reply 155

LeeMrLee

Original post by Jkn

Sorry guys, that's probably my bad. I set a good chunk of the problems on that thread (people are slacking :mad:

) and I've been getting really into integrals lately. :colone:

haha no its ok I have the step megapack and the smp further maths series to be getting on with and I need to teach myself m4/m5 so I'll be in a better position to do some harder stuff after xD

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Reply 156

Tedward

yeah the difficulty of question has seemed to escalate quickly. I guess its to be expected.

although what would be great for us alevel peeps is if you attached links to where we could learn the content to answer any uni level questions...

Reply 157

Felix Felicis

Original post by Jkn

You're going to need more detail to show you you got from your first line to your second. :tongue:

Edited my post :P

Also in reply to your last message :tongue:

... (sorry, didn't want to derail TPIT :P)

Original post by Jkn

Hmm, I suppose so (derive it from the general theorem using residues if you are feeling brave?). You are still king to have to do something with that ghastly limit definition though! :biggrin:

Haha oh dear, there was a fundamental flaw to my plan, lost sight of the original problem :tongue:

Ah, that's am extremely good shortlist! What was your reason for discarding Trinity then? :tongue:

It's just too grandiose for me :tongue:

I pretty much almost immediately crossed out Trinity and John's for that :lol:

I swear you've done loads! :lol:

You best do, you should feel like a dick! I'm pretty sure I have posted at least a fifth of the problems on this thread.. :lol:

Gah, I do

Sorry, I've been surprisingly busy lately, I haven't had that much time for maths actually :confused:

I've still only really looked at Laplace transforms and the Beta and Gamma functions so far this summer.

Very true. The proof of the Gamma-Beta link though.. :| (beautiful theorem no doubt!)

Certainly! I wrote a little note to the examiner about the wide applicability of what we had proven! :smile:

Haha

That's awesome :biggrin:

We had a discussion a while on the A level maths thread about writing messages to examiners - on one of the STEP I questions this year (which I didn't even attempt) I wrote a little message saying "I have discovered a truly marvelous proof of this, which this margin is too small to contain" :P

I hope so! :colone:

Not tempted to join?

I may do some time in the future! But as I've said, I've been surprisingly busy lately and haven't had that much time for maths yet but that'll change soon :biggrin:

They literally are! It's one of the few examples I know of of 'pure' problem solving (by which I mean problems that are easily stated in a non-ambiguous standard way, etc..). Perhaps skills with things like that would be useful if went in to fractional calculus or something like that? :tongue:

Good luck, if you don't get through, pay to go in! You're going to love it when you find it :colone:

Ha, fractional calculus is still a mystery to me but I'll have a look :P Mm, I dunno about paying for BMO2 - maybe for BMO1 but if I didn't make it through out of merit, I don't really see the point in paying to get my ass kicked on a desk by a wad of paper :lol:

I'll just look at it at the comfort of my home :tongue:

Reply 158

LeeMrLee

Original post by Tedward

I've only just finished A level stuff... it's mostly practice and careful thinking outside the box.

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Reply 159

Jkn

Original post by 0x2a

Nope, still can't do them. :colondollar:

Don't worry though, I only learnt integration by parts like two months ago and I have eons till I need to prepare for STEP or anything close to it. :biggrin:

Original post by LeeMrLee

haha no its ok I have the step megapack and the smp further maths series to be getting on with and I need to teach myself m4/m5 so I'll be in a better position to do some harder stuff after xD

Good luck with STEP guys, try not to get too narrowly obsessed in it though (like many of us do)! :biggrin:

Original post by Tedward

LOL it did!

Well that's what I made "A Summer of Maths 2013" for, as many people get discouraged by people posting advanced unexplained solutions on these problem-solving threads. In fact, it ended up really pissed me off because there were a few people who would refuse to explain one line proofs even after page-long conversations requesting more and more detail each time.. you then end up releasing that their proof was actually 3 pages long, they just didn't write it down.. meaning it became about 'point-scoring' <.<

Anyway, on ASOM I made a few teaching-resourse-type posts when solving problems on their and others have contributed with similar posts. There is also a resource library I am gradually building up (going to be doubling its size over the next few days!) The trouble is that I can't work out whether or not anyone reads or benefits form them so its hard to motivate myself to do them /:

In addition, I have started trying to set problems on TPIT that link from A-level ideas. For example, look at the recent headache I have given 'Felix Felicis' about deriving results that require two different Gamma function representations (he need to prove the representations link which is very very hard! :lol:

HAHA!) but this does mean that people will be able to understand and enjoy the solution more. Hopefully you feel the same? :smile:

Original post by Felix Felicis

Edited my post :P

MULTIPLICATION THEOREM! The two words I was scanning your post for! Well done! (no idea how you thought you could get away with that :wink:

)

It's just too grandiose for me :tongue:

I pretty much almost immediately crossed out Trinity and John's for that :lol:

Good that you have a good idea of what you wanted! I originally was just aiming for elite and it took until visiting to realise that that is (probably?) not what I would have wanted. Also, I get the impression that, even if you starting doing kick-ass maths, that loads of the other students wouldn't give you any respect or think you have any worth unless you've done well in the IMO, TST or BMO2 or whatever (none of which I ever came close to qualifying for :lol:

) and I certainly don't want to be in that situation! Emmanuel seems as though it would be a nice fresh start for me. They let in the usual number of state school applicants and stuff like that and I get the impression that people would be treated according with their abilities rather than what achievements they already had (the interview attitude was certainly "we don't give a ****, show us what you can do right now on the spot!" which seemed as good an approach as any).

Don't think I would fare well at Warwick though! Since I found most of STEP quite easy (towards the end), I doubt I would be in good company there (and the STEP study days I went on certainly made me feel like the know-it-all at the back of the class in many instances). I want to meet people that make me think "****, they're amazing! I better get studying/practicing if there are people like that out there!".. like you! :lol:

Gah, I do

Sorry, I've been surprisingly busy lately, I haven't had that much time for maths actually :confused:

I've still only really looked at Laplace transforms and the Beta and Gamma functions so far this summer.

Better than nothing! :smile:

Haha

That's awesome :biggrin:

Hahaha I think that joke has gone way too far! Poor Fermat and his troubles with infinite descent :lol:

I cocked up STEP I majorly, how did you do? I thought I was on track for 6 full questions (taking my time on each one) and then suddenly, after my 5th, I realised I has like 10 minutes left to do some domain/range question which I probably cocked up on! :/

I'll just look at it at the comfort of my home :tongue:

I don't know why everyone is so cautious with it! When we think about multiplying by -1 we think of discrete changes flicking back and forth across an axis (or whatever)... but then we introduce fractional powers, irrational, transcendental and even imaginary powers. At first we cannot give meaning to

(-1)^{\pi}

because we think "how can we change the sign of a number a non-integer number of times".. but then we realise that, upon introducing i, we can get the value in terms of sine and cosine immediately! :smile:

Meh I'm sure you'll destroy it anyway! :tongue: