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The 2013 STEP Prep Thread!

Prove there exists no polynomial

f(x)

with integer coefficients such that

f(a)=b, \ f(b)=c, \ f(c)=a

for

a, b, c

distinct integers.

Reply 1781

und

16

Original post by DJMayes

Well, all I can say is I'm glad you're not applying to the same college at Cambridge as I am :tongue:

Half of my competition at St John's probably come from about three public schools so I'm not sure about applying to my college either. :rolleyes:

Reply 1782

username937760

16

Original post by und

Half of my competition at St John's probably come from about three public schools so I'm not sure about applying to my college either. :rolleyes:

You're not going to be in the TSR Pembroke Maths war though :wink:

Reply 1783

und

16

Original post by DJMayes

You're not going to be in the TSR Pembroke Maths war though :wink:

I'm guessing you find yourself embroiled in that then. I have a feeling that the Pembroke admissions tutors are going to be pleasantly surprised this year.

Reply 1784

_Georgie

Original post by und

I'm guessing you find yourself embroiled in that then. I have a feeling that the Pembroke admissions tutors are going to be pleasantly surprised this year.

Before they start stalking us on TSR

Reply 1785

_Georgie

Original post by Cephalus

For functions like 1/(1+x), the maclaurin expansion is the same as its binomial expansion, which is what I think they are expecting

Ah, I was thinking binomial but it didn't quite work. I'll take another look tomorrow when my eyes can focus again :tongue:

Reply 1786

username937760

16

Original post by und

I'm guessing you find yourself embroiled in that then. I have a feeling that the Pembroke admissions tutors are going to be pleasantly surprised this year.

Yeah, I've applied there. It will be interesting to see how that plays itself out :tongue:

Reply 1787

Dog4444

10

Original post by L'art pour l'art

Prove there exists no polynomial

f(x)

with integer coefficients such that

f(a)=b, \ f(b)=c, \ f(c)=a

for

a, b, c

distinct integers.

Is it NOT from Putnam or IMO or something similar in difficulty? Just deciding if I should try it. :biggrin:

Reply 1788

L'art pour l'art

14

Original post by Dog4444

Is it NOT from Putnam or IMO or something similar in difficulty? Just deciding if I should try it. :biggrin:

Got it from someone who said it's from a book on the theory of functions. Give it ago. :wink:

Reply 1789

TheMagicMan

15

Original post by Lord of the Flies

If you want weird take a look at

x^{x^{x^{\cdots}}}

Determine the interval of convergence for

^{ \infty} x

Spoiler

Reply 1790

Lord of the Flies

OP

19

Original post by TheMagicMan

Determine the interval of convergence for

^{ \infty} x

Spoiler

Easy to show that the upper bound is

e^{\frac{1}{e}}

, but I am encountering a problem with oscillations for small x, which I am assuming is the difficulty in the problem - the issue can probably solved by separating cases of odd and even tetration, but given that it's 6:30 am I need to get some sleep first.

(I think we had only resolved part of the problem on the other thread - the easy part) :rolleyes:

Reply 1791

Oromis263

2

Pretty pleased with myself today, I managed 1,2,3 and 5 in an hour before I went to sleep last night of the 2006 STEP I paper.. Bit annoying as I've taken quite a break from practise :lol:

Also, I attempted to differentiate

x^{x^{x}}

and was wondering if anyone could confirm my answer.

Spoiler

and my attempt at

x^{x^{x^{x}}}

Spoiler

(edited 11 years ago)

Reply 1792

und

16

Original post by Oromis263

Pretty pleased with myself today, I managed 1,2,3 and 5 in an hour before I went to sleep last night of the 2006 STEP I paper.. Bit annoying as I've taken quite a break from practise :lol:

Also, I attempted to differentiate

x^{x^{x}}

and was wondering if anyone could confirm my answer.

Spoiler

and my attempt at

x^{x^{x^{x}}}

Spoiler

Stick it into Wolfram if you want to be sure but from a quick glance it looks much the same as my answer. It's easy to see what the pattern is in general.

Reply 1793

Zuzuzu

12

Original post by Oromis263

Pretty pleased with myself today, I managed 1,2,3 and 5 in an hour before I went to sleep last night of the 2006 STEP I paper.. Bit annoying as I've taken quite a break from practise :lol:

Also, I attempted to differentiate

x^{x^{x}}

and was wondering if anyone could confirm my answer.

Spoiler

and my attempt at

x^{x^{x^{x}}}

Spoiler

Yes, they're right.

Extension: Prove by induction a general formula for

\frac{\mathrm{d} }{\mathrm{d} x} ^{n}x

.

Reply 1794

und

16

I think we should have some easier interview style questions in this thread, like:

Find

\int^{1}_{-1} x^{-2}

Reply 1795

Oromis263

2

Original post by Zuzuzu

Yes, they're right.

Extension: Prove by induction a general formula for

\frac{\mathrm{d} }{\mathrm{d} x} ^{n}x

.

Will try it now. :smile:

Original post by und

I think we should have some easier interview style questions in this thread, like:

Find

\int^{1}_{-1} x^{-2}

It doesn't converge? :s-smilie:

Although I did notice it was even, and changed the limits in the hope it might help, sadly it didn't.

Reply 1796

Llewellyn

14

[QUOTE="Oromis263;40175826"]
It doesn't converge? :s-smilie:

Although I did notice it was even, and changed the limits in the hope it might help, sadly it didn't.Sketching the graph reveals all... I think it's almost always worth doing that in integration (unless the function would take a while to sketch).

A similar example may include: