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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#1** Solve:

a)

b)

**#2** Find all functions satisfying:

**#3** Find all functions satisfying:

**#4** Find all functions satisfying:

**#5** Let a be a given real number. Find all functions satisfying:

**#6** Find all injective functions satisfying:

*Last edited by Kurre (2009-05-05 05:21:26)*

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

Well I would like to change the topic to "kurres exercises" or something, but I cant, so here comes a few more that are not related to functional equations.

**#7** Let

Find expressed in A

**#8** Let H be a subgroup of G. prove that the following statements are equivalent:

a) H is a normal subgroup

b) for all a,b in G,

**#9** Solve the following equation in primes p,q,r:

*Last edited by Kurre (2009-05-06 02:07:52)*

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Changed! You can edit a topic's title by editing its first post.

Why did the vector cross the road?

It wanted to be normal.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

Thanks! But I still cant find a field where to edit the title? i know what u mean, i have seen it before. I thought it had been too long since i created the topic..

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Hmm. Maybe the software has changed then. I'm certain it used to be possible that way.

(It's how I changed the title, but I probably follow different rules)

Good point about my answer, I forgot about that detail. I can tweak it though!

Why did the vector cross the road?

It wanted to be normal.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

mathsyperson wrote:

Changed! You can edit a topic's title by editing its first post.

You can edit a thread title for something like the first five minutes after you start the thread; after that period, you cant edit the title any more (except for moderators).

I know, its hard for moderators to know all the problems that non-moderators go through.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Thanks for clarifying!

Why did the vector cross the road?

It wanted to be normal.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

correct mathsy

**hints**

*Last edited by Kurre (2009-05-06 02:20:42)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Why did the vector cross the road?

It wanted to be normal.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

nice

*Last edited by Kurre (2009-05-06 08:27:15)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#10** let f be a function from the natural numbers to the natural numbers satisfying

*Last edited by Kurre (2009-05-06 08:55:34)*

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

uhm okey that was much nicer than what I did

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#11** Find all functions from the positive rational numbers to the positive rational numbers satisfying:

*Last edited by Kurre (2009-05-07 05:28:36)*

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#12**

Let n be an integer with m prime factors (not necessarily distinct). suppose d|n and define

and determine when equality occurs.

*Last edited by Kurre (2009-05-08 00:15:34)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Hi Kurre;

I have been working very hard on your #7. I will continue to work on it, but in case I don't get there after a couple of weeks more. I would very much like to see your method.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

bobbym wrote:

Hi Kurre;

I have been working very hard on your #7. I will continue to work on it, but in case I don't get there after a couple of weeks more. I would very much like to see your method.

Im really glad to hear you are working on it

If you want, I can post my solution.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#13** Let M be a (p-1)x(p-1) matrix where p is an odd prime number and each row has each element from {1,2,3,...,p-2,p-1} exactly once. Prove that p|det(M)

*Last edited by Kurre (2009-05-15 09:13:12)*

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

#13 is false for p=2.

Why did the vector cross the road?

It wanted to be normal.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

mathsyperson wrote:

#13 is false for p=2.

true, forgot about that. Fixed!

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

can be generalized. For each integer , divides the determinant of a matrix in which each positive integer less than and coprime with appears exactly once in each row and in each column.

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

JaneFairfax wrote:

can be generalized. For each integer , divides the determinant of a matrix in which each positive integer less than and coprime with appears exactly once in each row and in each column.

Yea I realized that when trying to sleep yesterday

edit: you dont need exactly once in the columns also

*Last edited by Kurre (2009-05-15 23:18:45)*

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

**#14**

let

?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Incomplete answer so everyone else please continue to look

*Last edited by bobbym (2009-05-16 11:11:55)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

**Online**