A Summer of Maths

OP

Original post by TheMagicMan

Highly... I think they're significantly better than the burkhill books which are the most common analysis books

Added to the OP.

Reply 182

OP

Original post by Blutooth

...

Modular arithmetic won. I'd rather explain it to a secondary school student. :biggrin:

Question:

If

x

and

y

are real numbers with

y \geq 0

and

y(y + 1) \leq (x + 1)^2

, then the inequality

y(y-1) \leq x^2

is satisfied.

Prove or disprove the above statement.

Reply 183

Original post by jack.hadamard

Modular arithmetic won. I'd rather explain it to a secondary school student. :biggrin:

Question:

If

x

and

y

are real numbers with

y \geq 0

and

y(y + 1) \leq (x + 1)^2

, then the inequality

y(y-1) \leq x^2

is satisfied.

Prove or disprove the above statement.

lol, so you're onto more interesting questions then :smile:

Reply 184

OP

Original post by Blutooth

lol, so you're onto more interesting questions then :smile:

Well, the interesting part of this question was to find an elementary solution; I couldn't see one.
Otherwise, it is a question like any other, but if you have suggestions I can still chew a bit on it.

Reply 185

OP

Original post by Lord of the Flies

Question

Find

f:\mathbb{R}\to\mathbb{R}

such that

\forall (x,y)\in\mathbb{R}:

\displaystyle f \left( x^2+f(y) \right)=x f(x)+y

Just started with this one.

Spoiler

Reply 186

Original post by Lord of the Flies

Question

Find

f:\mathbb{R}\to\mathbb{R}

such that

\forall (x,y)\in\mathbb{R}:

\displaystyle f \left( x^2+f(y) \right)=x f(x)+y

Spoiler

(edited 11 years ago)

Reply 187

Lord of the Flies

19

Original post by jack.hadamard

Just started with this one.

Spoiler

There are two - and yes, the involution is key.

Reply 188

shamika

Original post by jack.hadamard

I see. Well, I will focus on Real Analysis for the moment; it doesn't quite suit me.

Do you recommend these Zorich's books, so that I add them in the OP?

Yes - they are probably the best analysis books I've come across. Annoyed I didn't know about them when I was an undergrad

Reply 189

TheMagicMan

15

Original post by Lord of the Flies

There are two - and yes, the involution is key.

Yeah I got that there are two... Beyond an inductive approach is there a simple solution?

This was posted from The Student Room's iPhone/iPad App

Reply 190

Does anyone on here like combinatorial problems cos I've got a shedload of those. Also got quite a few olympiad style number theory if anyone is interested.

Reply 191

Original post by TheMagicMan

Yeah I got that there are two... Beyond an inductive approach is there a simple solution?

This was posted from The Student Room's iPhone/iPad App

I did not know what involution was and still used it. Seems a relatively simple technique to me.

Edit: misread induction as involution \facepalm :P

(edited 11 years ago)

Reply 192

boromir9111

17

Original post by jack.hadamard

Do you recommend these Zorich's books, so that I add them in the OP?

Reply 193

OP

Original post by Blutooth

Damn you beat me to it. Oh and there is another thing f(x) could be.

Yeah, just realised; I was more or less guessing.

Original post by Lord of the Flies

There are two - and yes, the involution is key.

That is a nice question then. Do you have some more similar? :tongue:

Reply 194

OP

Original post by Blutooth

Does anyone on here like combinatorial problems cos I've got a shedload of those.

I generally like to bang my head in the wall, but it depends on whether they require undergrad combinatorics. :biggrin:

Reply 195

Lord of the Flies

19

Original post by TheMagicMan

Yeah I got that there are two... Beyond an inductive approach is there a simple solution?

This was posted from The Student Room's iPhone/iPad App

Yes there is a fairly simple solution!

EDIT: in fact Blutooth posted it a few minutes ago! :biggrin:

(edited 11 years ago)

Reply 196