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# The Proof is Trivial! watch

1. Solution 236

Here's a more systematic way:

Ditto for .
2. (Original post by Jkn)
Edit: Another fun problem (not from IMO):

Problem 234*

Find all possible n-tuples of reals such that and for
Perhaps add an explanation of the Pi notation and tuples, since they're not mentioned at A-level (or not on Edexcel, at least) - I know it's just notation, but when you don't recognise words/symbols it can put you off pretty quickly.
3. (Original post by The Polymath)
Perhaps add an explanation of the Pi notation and tuples, since they're not mentioned at A-level (or not on Edexcel, at least) - I know it's just notation, but when you don't recognise words/symbols it can put you off pretty quickly.
Perhaps notation should get a note in the OP?
4. (Original post by The Polymath)

Spoiler:
Show

Solution 238
I may be simplifying things too much here, but would it just be ?
Yeah, that's it baby girl. How'd you solve it?
5. Solution 237

Let

6. (Original post by hecandothatfromran)
Yeah, that's it baby girl. How'd you solve it?
7. (Original post by bananarama2)
Yeah, i know . That's how i talk. To me, it has an aura of eloquence about it, don't you think?
8. (Original post by The Polymath)
...
Did you quote me? It seems so, but I can't find it.
9. (Original post by Felix Felicis)
Solution 237

Let

Yep, but it can be simplified further.
10. (Original post by Jkn)
Had a nice afternoon avoiding STEP and indulging in some IMO problems!
Did you not like problem 6? Determine whether is a perfect cube for some .
11. (Original post by james22)
Yep, but it can be simplified further.
So it can
12. (Original post by bananarama2)

http://cdn.memegenerator.net/instanc...0/38781038.jpg
13. (Original post by The Polymath)
Perhaps add an explanation of the Pi notation and tuples, since they're not mentioned at A-level (or not on Edexcel, at least) - I know it's just notation, but when you don't recognise words/symbols it can put you off pretty quickly.
(Original post by bananarama2)
Perhaps notation should get a note in the OP?
I think everyone knows at at this point, perhaps there should be some basic notation and even teaching resources in the OP, that would be great...

OMG I've just had a vision... like a TSR olympiad society where everyone learns random **** and then some of the older people find 6 or so problems and put them on and then we all have to do them in time conditions... that would be awesome! :O

I was a bit apprehensive in typing the problem up in that form though, without that notation, it is far too obvious to spot (I shall say no more )

Also, I actually typed 10 problems up (rather than 4) and yet my browser froze and TSR only saved the first 4 :/ I'll probably put them up later...
Did you not like problem 6? Determine whether is a perfect cube for some .
I gave it a minute or so but, as nothing jumped out at me, I decided not to try it (I don't have enough experience with functional equations to know whether or I'm on the right track...)

Spoiler:
Show
Have you done Problems 2 and 3? If so, were you delighted with what the solutions turned our to be?
Did you quote me? It seems so, but I can't find it.
Yeah, a response to your question, but I realised that LotF had already given the answer so I deleted it. That quote notification is doomed to remain a the top of your "who quoted me" for the rest of time (really annoying bug)
15. (Original post by The Polymath)
Yeah, a response to your question, but I realised that LotF had already given the answer so I deleted it. That quote notification is doomed to remain a the top of your "who quoted me" for the rest of time (really annoying bug)
Okay, I was just puzzled.

(Original post by Jkn)
I gave it a minute or so but, as nothing jumped out at me, I decided not to try it (I don't have enough experience with functional equations to know whether or I'm on the right track...)

Spoiler:
Show
Have you done Problems 2 and 3? If so, were you delighted with what the solutions turned our to be?
This is a wise decision. Well, I have been out of practice (done only bookwork) for a while and feel atrophied already. I did problem two easily, because I have been using Vandermonde-related identities recently. About problem three, I recognised Cassini's identity and got the right answer, but through a flawed proof. Problem six is actually a breeze.
Spoiler:
Show

This is a wise decision. Well, I have been out of practice (done only bookwork) for a while and feel atrophied already. I did problem two easily, because I have been using Vandermonde-related identities recently. About problem three, I recognised Cassini's identity and got the right answer, but through a flawed proof. Problem six is actually a breeze.
Spoiler:
Show

Did you prove the identity? I think you would have needed too! I was desperately trying to force an elegant solution... and finally got one Working backward you can see how simply the massive ugly summation is going to end up so, by using combinatorial methods, you can actually (more or less) derive the formula directly! Oh and I also attempted induction to try and have an alternate solution but it got waaay too grizzly!

As for problem 3, what's Cassini's identity? Oh right hmm Me and Zakee had a go at that one together. After a while, I realised I could simplify it be using the Pell equation (where I only needed to calculate one of the values) and so the solution, albeit tediously, came out. A while later we ended up finding the elegant solution which is a real mind-blower! I would give you more detail but I don't want people to lose interest in the problem who are undoubtedly reading this despite the spoiler
17. Solution 232

Clearly the possible elements are from the set and, for each of those guys, we have corresponding subsets. Hence, .

Solution 233

We claim that all solutions to belong to the set consisting only of consecutive Fibonacci numbers.
Suppose that there is a solution , which is not in . We can further suppose that is minimal and that . Thus is a solution with smaller sum; whence, it is in , implying that is in - contradiction.

Solution 234

Really?!

Solution 235

I denote the left handed side by , for I am too lazy to type this sum. For the inequality holds true. Suppose that it is true for all .
Then, .
It is quite weak.
18. (Original post by ukdragon37)
I can't believe it.... ten thousand words of category theory, done and submitted. I never thought I'd be able to

Now to get some sleep.
Congrats btw
19. (Original post by Jkn)
Cassini's identity is regularly used in STEP. You are meant to recognise that and prove no other solutions.

...
You are on the team this year, then? It's approaching.
Solution 232

Clearly the possible elements are from the set and, for each of those guys, we have corresponding subsets. Hence, .
Seen as it is a problem that gives you a result to prove, you should probably elaborate on your last step (as a rule of thumb, if an IMO examiner would not award you the marks, you can probably assume that we will have trouble following )
Solution 233

We claim that all solutions to belong to the set consisting only of Fibonacci numbers.
Suppose that there is a solution , which is not in . We can further suppose that is minimal and that . Thus is a solution with smaller sum; whence, it is in , implying that is in - contradiction.
This is incorrect. You have assumed a statement without justification and then discarded the 'counterexample' in favour of the conjecture. You have also not fully justified how it is that you have a contradiction and you have also applied an algebraic manipulation that contradicts the generality of a result you later used.
Solution 234

Really?!
If you find a problem trivial, I don't see why you need to put other people off. I'm sure someone would have enjoyed posting a solution to it :/
Spoiler:
Show
I think posting in a problem like this every now and then is good. The notation will mean that many will likely faff about with formulae for ages and so learn a valuable lesson

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