# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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## #1 2009-04-17 23:33:01

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

### Kurre's Exercises

#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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## #2 2009-05-04 06:45:12

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

### Re: Kurre's Exercises

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,

if and only if

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

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

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## #3 2009-05-04 08:50:47

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

### Re: Kurre's Exercises

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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## #4 2009-05-04 10:30:08

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

### Re: Kurre's Exercises

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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## #5 2009-05-04 11:03:26

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

### Re: Kurre's Exercises

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)

Why did the vector cross the road?
It wanted to be normal.

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## #6 2009-05-04 20:50:17

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

### Re: Kurre's Exercises

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 cant edit the title any more (except for moderators).

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

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## #7 2009-05-04 23:29:08

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

### Re: Kurre's Exercises

Thanks for clarifying!

Why did the vector cross the road?
It wanted to be normal.

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## #8 2009-05-06 02:18:20

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

### Re: Kurre's Exercises

correct mathsy

hints

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

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## #9 2009-05-06 06:16:44

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

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## #10 2009-05-06 06:54:14

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

### Re: Kurre's Exercises

Why did the vector cross the road?
It wanted to be normal.

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## #11 2009-05-06 08:19:49

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

### Re: Kurre's Exercises

nice

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

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## #12 2009-05-06 08:27:55

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

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## #13 2009-05-06 08:52:51

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

### Re: Kurre's Exercises

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

if n-1>m>0. Find the least possible value of f(2009) and f(2011)

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

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## #14 2009-05-06 09:07:40

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

### Re: Kurre's Exercises

uhm okey that was much nicer than what I did

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## #15 2009-05-07 05:18:05

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

### Re: Kurre's Exercises

#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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## #16 2009-05-08 00:15:18

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

### Re: Kurre's Exercises

#12
Let n be an integer with m prime factors (not necessarily distinct). suppose d|n and define

where
and
are the multiplicities for p in the prime factorization for n and d respectively. Show that:

and determine when equality occurs.

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

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## #17 2009-05-12 03:33:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Kurre's Exercises

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #18 2009-05-15 08:12:30

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

### Re: Kurre's Exercises

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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## #19 2009-05-15 08:26:45

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

### Re: Kurre's Exercises

#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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## #20 2009-05-15 09:07:22

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

### Re: Kurre's Exercises

#13 is false for p=2.

Why did the vector cross the road?
It wanted to be normal.

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## #21 2009-05-15 09:15:43

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

### Re: Kurre's Exercises

mathsyperson wrote:

#13 is false for p=2.

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## #22 2009-05-15 19:46:48

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

### Re: Kurre's Exercises

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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## #23 2009-05-15 23:15:07

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

### Re: Kurre's Exercises

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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## #24 2009-05-16 02:48:54

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

### Re: Kurre's Exercises

#14
let

be the n roots of unity. For which n and m does it hold that:
?

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## #25 2009-05-16 11:05:43

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606