I don't know really. It didn't feel too imposing and the people were really friendly when I went to look. The ducks were a big attraction too . I have to say I'm not interested in the REALLY pure stuff which crops up here, but I do like a good maths puzzle.(Original post by Jkn)
Awesome, what made you choose emmanuel specifically? Oh thanks Same goes for you! I know few science students interested in pure maths (though I myself have a great interest in theoretical physics)
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bananarama2
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 26042013 15:30

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 26042013 16:33
(Original post by Mladenov)
No, you are not allowed to attend university. I have done all IMO SL problems from 1990 to 2010 (with several exceptions of course  see, for instance, SL 2003, A6). I am planning to do SL 2011 as a mock before our TST. The Iranian TST is very challenging, I would say it is more difficult than the Chinese TST. Another good source of problems is the AllRussian MO (especially combinatorics and geometry). For inequalities  Vietnamese TST; number theory  almost everywhere(I am biased here). The French TST is somewhat tough. Our NMO and TST are also not easy. I can post some problems if you want. But just look at, for example, the Italian TST, or the German TST  these are warmups. I will not even mention Spain, Portugal, and the rest. In Europe, broadly speaking, olympiad mathematics is abandoned.
In my country there are high schools which emphasize on olympiad mathematics and this makes the difference. The students at these schools study, generally speaking, mathematics and languages.
The best students sometimes work with professors.
I cannot not understand your argument. Could you please explain?
Your approach makes me think of the density of sequences of integers.
(Reading through this part does seem horribly sloppy!) By successive terms equaling integers I meant that the difference between n and was always as integer. I lack the tools to explain what I mean rigorously but I thought the general idea seemed fairly intuitive:
Consider the values of the sequence where supposedly "passes by" an integer r. As the sequence can only ever increase by fractions (bounded by an upperbound that is inversely proportional to as n approaches infinity  and also fractions) this would require successive terms of to equal two rational numbers above and below which would, in turn, imply that (not coprime, my bad!) for successive terms of n as n moves past r. This is a contradiction as, to "pass by" the integer would clearly require divisibility for one of the adjacent terms (i.e. r).
I'll attempt to formalise the last bit: clearly as it "passes by" the value of r. So let . This implies which implies is not an integer (as its between n and n+1) therefore the inequality is not strict and r does infact equal either of the two (i.e. is in ). This is contrary to our assumption. Therefore no such r exists. QE"mother****ing"D.
Excellent. I would have never even thought that this might be the case.
Regarding the geometric interpretation  I also suspected something, but AMGM seemed more reliable.Last edited by Jkn; 26042013 at 17:56. 
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 26042013 16:41
(Original post by bananarama2)
I don't know really. It didn't feel too imposing and the people were really friendly when I went to look. The ducks were a big attraction too . I have to say I'm not interested in the REALLY pure stuff which crops up here, but I do like a good maths puzzle.
It's literally such a small world isn't it! Are you considering the masters in E&T.Phys? 
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 26042013 16:41
Problem 82**
We have not done any combinatorics.
Let and are such that:
i) each element can be represented as , where ;
ii) if and , then .
Then, we have .
Problem 83**
Let , , and be positive integers such that . Then and . 
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 26042013 16:59
(Original post by Jkn)
Mm I couldn't agree more. I went to Trinity on my open day and, although it was nice to see the setting to a lot of books I read, I realised it wasn't for me (packed with tourists, too big, admissions process too biased towards private schools, etc, etc...) Emmanuel on the other hand Even so, it's nice to have physicists with a general interest in mathematics itself.
It's literally such a small world isn't it! Are you considering the masters in E&T.Phys?
I actually applied to do Chemical Engineering in my second year, but I have a choice and I can't choose between Physics and Chem Eng. 
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 26042013 17:13
(Original post by bananarama2)
Even at interview Emma seemed inviting and friendly. Well Physics is written in maths
I actually applied to do Chemical Engineering in my second year, but I have a choice and I can't choose between Physics and Chem Eng.
Tell that to my physics teacher, he having none of it! I don't think he understands tautologies either The other day we were were doing radioactivity equations for medicine and he said something ridiculous like the fundamental things that are conserved are "Energymass and baryon number" and I was like "well what's baryon number then it's a concept independent of both energy and mass?" and he took a big sigh and was like "*facepalm* baryon number is a third" ?!?!?! :L
I was actually planning to apply for physics until halfway through last year! 
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 26042013 17:25
(Original post by Jkn)
Talking to other people I met at the STEP residential I went on over easter, the general consensus was that ours was fairly brutal! I think everyone came out shaking :L Whilst I was there though our DOS came to see us though (which I don't think he was supposed to!) and somehow he remembered me
Tell that to my physics teacher, he having none of it! I don't think he understands tautologies either The other day we were were doing radioactivity equations for medicine and he said something ridiculous like the fundamental things that are conserved are "Energymass and baryon number" and I was like "well what's baryon number then it's a concept independent of both energy and mass?" and he took a big sigh and was like "*facepalm* baryon number is a third" ?!?!?! :L
I was actually planning to apply for physics until halfway through last year!
Why did you change your mind? 
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 26042013 17:33
(Original post by Jkn)
Is the "IMO SL" the IMO? ;0 Haven't been able to find any TSTs (except the UK's) but they seem pretty beyond me, for now anyway, I think I'd be lucky to find one I could do amongst 5 papers! None the less it would be nice to see some of the reopen TSTs if they're a little easier (perhaps you could quote the sources) so post away! Also a few BMO1/BMO2level questions would be appreciated
Spoiler:Show
(Original post by Jkn)
Hmm that sounds interesting, is that what you do? Do you have to pay? In my country we have schools like that that cost tens of thousands of pounds (or more!) per year to attend (though they are nonspecialist) but in a few years they are introducing maths academies, similar to what you have now perhaps, that train students in problem solving and train students for STEP in the same way students are coached towards ALevels.
(Original post by Jkn)
When I said when the sequence could decrease I was aiming to account for the cases where (and aimed to note that tho was irrelevant).
(Reading through this part does seem horribly sloppy!) By successive terms equaling integers I meant that the difference between n and was always as integer. I lack the tools to explain what I mean rigorously but I thought the general idea seemed fairly intuitive:
Consider the values of the sequence where supposedly "passes by" an integer r. As the sequence can only ever increase by fractions (bounded by an upperbound that is inversely proportional to as n approaches infinity  and also fractions) this would require successive terms of to equal two rational numbers above and below which would, in turn, imply that (not coprime, my bad!) for successive terms of n as n moves past r. This is a contradiction as, to "pass by" the integer would clearly require divisibility for one of the adjacent terms (i.e. r).
I'll attempt to formalise the last bit: clearly as it "passes by" the value of r. So let . This implies which implies is not an integer (as its between n and n+1) therefore the inequality is not strict and r does infact equal either of the two (i.e. is in ). This is contrary to our assumption. Therefore no such r exists. QE"mother****ing"D.
(Original post by Jkn)
It was staring us in the face the whole time! AMGM is overkill in so many situations! Plus you don't tend to learn anything new about the equation. I see it as a subsidiary 'implicit' method, just like induction (though, of course, these methods are highly useful if all you need to do is 'prove' or 'solve'), wouldn't you agree? 
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 26042013 17:56
(Original post by bananarama2)
My maths interview was alright, although I blanked and I blanked on a chemistry question in my science interview too. I'd met my interviewer at the open day but I don't think he remembered me. Teachers do have a habit of saying what they just said in a slightly different way when you want more detailed information.
Why did you change your mind?
Well I was only every really interested in theoretical physics and 'the big questions' (sheldoncooper style, standard) and I met some current student at my open day, one from maths and one from natsci, and they said that maths was 100% the best route into it and that even 'maths with physics' in my first year would hold me back!
That was the first step and since then I've found that the beauty and satisfaction in pure maths is immense and is completely different to everything else I've experienced. Besides, I'm not really fussed about money and I don't want to dedicate my life to coming up with things that makes peoples lives easier, so why not explore the cosmos/ mathematical universe!
(Original post by Mladenov)
SL=Shortlist
Spoiler:Show
Yes, it is what I do. Nope, we do not have to pay(these schools are public, as almost everything here).
Right, I see. Now, take it as an advice, not a pedantic remark, you have to prove that for some positive integer in order your solution to be complete.
I agree, of course. It is a nice idea to delve into the problem, but this is possible only when you have time and interest in the problem.
Oh thats awesome
But that's fairly trivial because is r=1, rather than n, then 
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 26042013 18:35
(Original post by Jkn)
Cheers! Your MO 3rd round looks bizarrely accessible :O
Oh thats awesome
But that's fairly trivial because is r=1, rather than n, then
Our third round problems are for the daycare.
Try 4th round problems or TST. 
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 26042013 18:40
(Original post by Mladenov)
The idea was to secure the existence of such that .
Our third round problems are for the daycare.
Try 4th round problems or TST.
I think we all have a higher level of rigour when writing on paper rather than typing latex code into a computer! Someone needs to invent a mathematical keyboard
If you know of some nice algebra ones then post away! 
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 26042013 18:59
(Original post by Jkn)
*coughcough*pedantic*coughcough* !!!
I think we all have a higher level of rigour when writing on paper rather than typing latex code into a computer! Someone needs to invent a mathematical keyboard
If you know of some nice algebra ones then post away!
That is why I said my remark was rather punctual.
What kind of algebra?
Here is one interesting.
Problem 84**
Let be a set of elements. Denote by the set of all subsets of .
Suppose satisfies:
i) for every subset of , we have ;
ii) for every two subsets and of , we have .
Then, the number of positive integers such that there exists with is less than .Last edited by Mladenov; 26042013 at 19:02. 
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 26042013 19:03
(Original post by Mladenov)
What kind of algebra?
Having said that it is nice with problems like 80 where we collective produce 4 radically different, yet equally valid, solutionsLast edited by Jkn; 26042013 at 19:04. 
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 26042013 19:14
(Original post by Jkn)
Not necessarily algebra, just one */** problems. I haven't done set theory, amgm, etc... formally and so my understanding of them is likely to be very limited (and most people on here can probably empathise). (though it is nice to 'have a go')
Having said that it is nice with problems like 80 where we collective produce 4 radically different, yet equally valid, solutions
I am not acquainted with the A level content, so I hope that this is suitable.
Problem 85*
How many common terms have the following arithmetic progressions:
and , if they have the same number of terms  ?
Problem 86*
Solve in integers:
, where is even number. 
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 26042013 19:41
(Original post by Mladenov)
Problem 85*
How many common terms have the following arithmetic progressions:
and , if they have the same number of terms  ?
Sequences are for nonnegative integers n and m. As each term has at most 121 terms so the largest common terms is .
Equating the two sequences gives so . This implies common sequence= for some nonnegative integer p. As so .
Therefore there are 25 solutions to the original equations.
(really mladenov?) 
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 26042013 20:00
Let for some positive integer m.
so y is odd. Solving the quadratic gives... and, as y is odd this is an integer if and and if for some integer q.
Rearranging we get . As 3 is prime the possible factors are (±1,±3), (±3,±1).
This gives ±1 (taking note of the parity of m) and hence is the only solution.
Substituting back in gives . Therefore all possible solutions are:
(x,y)=(0,±1) or (1,±1)Last edited by Jkn; 29042013 at 09:58. 
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 26042013 20:03
Nice. I will give you something different.
Check your solutions to problem 86. You have missed solutions and is not a solution, for example.
Problem 87*
A group of boys and girls went to a dance party. It is known that for every pair of boys, there are exactly two girls who danced with both of them; and for every pair of girls there are exactly two boys who danced with both of them. Prove that the numbers of girls and boys are equal.Last edited by Mladenov; 26042013 at 20:07. 
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 26042013 20:18
(Original post by Mladenov)
Nice. I will give you something different.
Check your solutions to problem 86. You have missed solutions and is not a solution, for example.
I'll give it a go when I get some time!Last edited by Jkn; 26042013 at 20:21. 
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 26042013 20:27
Spoiler:Show
since is even we can write for an integer so
rearanging gives
since 3 is prime and both brakets are integers we can consider 4 sets of simultanious equations (only possibilities for the factors are (1,3), (3,1), (1,3), (3,1))
solving these gives x=0 or x=1
plugging these into the original equation gives solutions
hence these are the only solutions.
Last edited by james22; 26042013 at 21:01. 
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 27042013 16:44
Just remembered a really nice STEP question I did while ago and thought of this...
Problem 88*
is defined as the largest product that can be made from positive integers that add up to N.
i.e. .
Prove the Goldbach Conjecture
Problem 89*
Find the smallest prime number such that the following is true:
The largest prime factor of N1 is A;
The largest prime factor of A1 is B;
The largest prime factor of B1 is 7.Last edited by Jkn; 27042013 at 16:49.
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