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# The Proof is Trivial! watch

1. (Original post by Smaug123)
I am sad that there doesn't seem to be a nice proof of this
Oh there definitely is. In fact it is one of the cleverest proofs I know, and it is about 3 lines long.
2. (Original post by Lord of the Flies)
Oh there definitely is. In fact it is one of the cleverest proofs I know, and it is about 3 lines long.
There's an infinite-descent proof of the form "suppose this fails; then we prove that there is an infinite number of points", but I don't think that proof is very nice do you have a different one in mind?
3. Problem 562*

Factorise as a product of real linear and quadratic polynomials.
Write as a product of real quadratic polynomials.
4. Solution 562
Spoiler:
Show

The roots of are where .

Hence we can write

For the final part note that hence

5. (Original post by Smaug123)
There's an infinite-descent proof of the form "suppose this fails; then we prove that there is an infinite number of points", but I don't think that proof is very nice do you have a different one in mind?
I think we are referring to the same idea then, though in your version contradiction/descent seems like a superfluous addition. It is more "do X, if A: end, if B: do Y and restart with X. There are finitely many points, this must halt, done".

I personally think the idea is extremely clever. Granted drawing lines isn't too pretty, but the idea is very hard to come by from the statement - not to mention that by nature of the problem, it feels overoptimistic to expect a proof that does not draw lines and look at properties of said lines. Different tastes I guess!
6. (Original post by Lord of the Flies)
I think we are referring to the same idea then, though in your version contradiction/descent seems like a superfluous addition. It is more "do X, if A: end, if B: do Y and restart with X. There are finitely many points, this must halt, done".

I personally think the idea is extremely clever. Granted drawing lines isn't too pretty, but the idea is very hard to come by from the statement - not to mention that by nature of the problem, it feels overoptimistic to expect a proof that does not draw lines and look at properties of said lines. Different tastes I guess!
whats the proof then
7. (Original post by newblood)
...
Spoiler:
Show
Pick any 2 points and draw a line through them: suppose a 3rd point lies on the line (else we are done) and pick the closest point to the line, at a distance say.

Of our 3 points a pair lies on one side of draw a line through and the furthest of the pair from . The distance between and the second point is

It is surprising, at least to me, that this statement was only first proved in 1943/44.
8. This **** crayyyyyyyyy
9. Problem 563*

Define a family of polynomials by:

,

Prove that and find the coefficient of in

Form and prove a conjecture about the significance of

Problem 564*

Suppose and are two nth order polynomials such that at n+1 distinct real values of x. Prove that .
10. (Original post by 16Characters....)
Problem 563*

Define a family of polynomials by:

,

Prove that and find the coefficient of in

Form and prove a conjecture about the significance of

Problem 564*

Suppose and are two nth order polynomials such that at n+1 distinct real values of x. Prove that .
I always thought the Chebysheff polynomials woukld be a good theme for a STEP question; they certainly have a lot of odd properties.

564 looks like another good 'not obvious unless you can prove it' type of question. I'll try both of these today.

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11. (Original post by Krollo)
I always thought the Chebysheff polynomials woukld be a good theme for a STEP question; they certainly have a lot of odd properties.

564 looks like another good 'not obvious unless you can prove it' type of question. I'll try both of these today.

Posted from TSR Mobile
Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.
12. (Original post by 16Characters....)
Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.
Have you done the question on bernoulli polynomials (87/3/10)?

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13. (Original post by 16Characters....)
Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.
There's a small section on orthogonal polynomials in some tripos courses, such as methods and numerical analysis.
The questions are a little limited though, but if you're interested it may be worth a look.
14. (Original post by Krollo)
Have you done the question on bernoulli polynomials (87/3/10)?

Posted from TSR Mobile
Nope, barely touched any of the pre-1993 papers yet but I'll have a look later in the week thanks.

(Original post by joostan)
There's a small section on orthogonal polynomials in some tripos courses, such as methods and numerical analysis.
The questions are a little limited though, but if you're interested it may be worth a look.
Thanks for the suggestion.
15. Solution 563*
(Please let me know of mistakes/improvements, Im only a 'learner'!). I also apologise for the lack of LaTeX as I wasnt planning on sharing it.

Maybe it was a bit of an overkill for the first part, but Im never too sure when it says to prove something which seems obvious.

Problem 565*
By considering the results from Problem 563, show that
16. Solution 565
Try Q7 here for a similar question.
Spoiler:
Show

Let

Letting :

Considering the case :

As required.

Now considering

As required

Now for

As required.

Note: I had a string of annoying LaTeX issues so did a lot of cutting and retyping. If it looks like something is missing, it probably is!
17. (Original post by 16Characters....)
Solution 565
Try Q7 here for a similar question.
Spoiler:
Show

Let

Letting :

Considering the case :

As required.

Now considering

As required

Now for

As required.

Note: I had a string of annoying LaTeX issues so did a lot of cutting and retyping. If it looks like something is missing, it probably is!
Lol yep i know what you mean, it it took a long while to type up the question to start with So are you a current student or graduate?
18. (Original post by EnglishMuon)
Lol yep i know what you mean, it it took a long while to type up the question to start with So are you a current student or graduate?
Year 13 lol, Joostan suggested I look at a Numerical Analysis worksheet because me and Krollo were discussing the Chebyshev polynomials. That question is the only one I am brave enough to even attempt!
19. (Original post by 16Characters....)
Year 13 lol, Joostan suggested I look at a Numerical Analysis worksheet because me and Krollo were discussing the Chebyshev polynomials. That question is the only one I am brave enough to even attempt!
Ah nice! Do you have any links to these worksheets? I've been having great fun on these polynomial questions. I first thought they were Bernoulli polynomials but it just turns out they have many cool properties in common. But yeah I should've know u were year 13, seen u around in the camb. Offers holders thread me thinks
20. (Original post by EnglishMuon)
Ah nice! Do you have any links to these worksheets? I've been having great fun on these polynomial questions. I first thought they were Bernoulli polynomials but it just turns out they have many cool properties in common. But yeah I should've know u were year 13, seen u around in the camb. Offers holders thread me thinks
Here are the Camb example sheets.

Also as Krollo recommended on the previous page, STEP III 1987 Q10 is about the Bernoulli poylnomials.

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