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#276 2005-12-14 17:26:26

Jai Ganesh
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Registered: 2005-06-28
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Re: Problems and Solutions

Problem # k + 63

Show that any prime number other than 2 can be expressed as the difference of two squares, where each square is an integer squared.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#277 2005-12-15 16:45:59

John E. Franklin
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Posts: 3,588

Re: Problems and Solutions


Last edited by John E. Franklin (2005-12-15 16:54:15)


igloo myrtilles fourmis

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#278 2005-12-15 17:11:58

Ricky
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Posts: 3,791

Re: Problems and Solutions


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#279 2005-12-16 20:58:20

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

I think Ricky is right. Here's a direct proof:
Let
u^2-v^2 = p
We'll find u and v by p:
u^2-v^2 = p
=> (u+v)(u-v) = p, but p is prime, so his divisors are only 1 and p, so:
u+v=p && u-v=1
=> u=v+1; 2v+1=p; v= (p-1)/2, but p is odd, so v is integer.
So:
v= (p-1)/2
u=1 + ((p-1)/2)= (p+1)/2.

Last edited by krassi_holmz (2005-12-16 21:00:23)


IPBLE:  Increasing Performance By Lowering Expectations.

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#280 2005-12-18 16:47:04

Jai Ganesh
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Posts: 48,425

Re: Problems and Solutions

Correct, John, Ricky and krassi_holmz!

Problem # k + 64

For what value of 'n' will the remainder of 351^n and 352^n be the same when divided by 7?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#281 2005-12-18 18:04:40

krassi_holmz
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Registered: 2005-12-02
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Re: Problems and Solutions

Last edited by krassi_holmz (2005-12-18 18:07:55)


IPBLE:  Increasing Performance By Lowering Expectations.

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#282 2005-12-18 22:58:39

Jai Ganesh
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Posts: 48,425

Re: Problems and Solutions

Correct! Well done, krassi_holmz smile

Problem # k + 65

The difference between a number and two-thirds of its value is the sum of the digits of the original number. How many digits can this number have?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#283 2005-12-19 00:13:34

mathsyperson
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Registered: 2005-06-22
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Re: Problems and Solutions

Last edited by mathsyperson (2005-12-19 00:13:44)


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

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#284 2005-12-19 05:30:29

Ricky
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Re: Problems and Solutions

Something I just noticed about k+63.  The same can be said for all odd numbers, no?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#285 2005-12-19 07:27:00

mathsyperson
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Re: Problems and Solutions

True.
The difference of 2 squares is (n+1)² - n² = n² + 2n + 1 - n² = 2n+1.
Odd numbers are written as 2n+1 by definition, so that proves it.


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

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#286 2005-12-19 16:21:47

Jai Ganesh
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Re: Problems and Solutions

Well done, Mathsyperson!

Problem # k + 66

If 5^m = 35^6 = 7^n, what is the value of (mn)/(m+n)?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#287 2005-12-20 16:08:41

Jai Ganesh
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Posts: 48,425

Re: Problems and Solutions

Problem # k + 67

The Arithmetic Mean of two numbers is 25 and their Geometric Mean is 20. What is their Harmonic Mean?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#288 2005-12-21 16:16:37

Jai Ganesh
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Re: Problems and Solutions

Problem # k + 68

A sports field consists of a rectangular region with semi-circular regions projecting at the two opposite sides. If the perimeter of the filed is 1,000 meters, find the area of the largest possible field.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#289 2005-12-21 17:59:25

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,425

Re: Problems and Solutions

Problem # k + 69

What is the least value of n such that n!>10^n?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#290 2005-12-22 02:20:27

mathsyperson
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Registered: 2005-06-22
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Re: Problems and Solutions

Last edited by mathsyperson (2005-12-22 02:20:37)


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

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#291 2005-12-22 14:16:24

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Re: Problems and Solutions

ganesh, do you have a list of the problems
that are unsolved?  That way I can go back
easier and solve some old ones.


igloo myrtilles fourmis

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#292 2005-12-22 16:41:32

Jai Ganesh
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Posts: 48,425

Re: Problems and Solutions

Well done, mathsyperson!
All of your solutions are correct.
The problem on AM, GM and HM was meant to be solved knowing AM*HM=GM²

Solution to Problem # k + 66

Problem : If 5^m = 35^6 = 7^n, what is the value of (mn)/(m+n)?

Solution : Let 5^m=(5*7)^6=7^n=k
5^m=k, therefore, 5 = k^(1/m)
7^n=k, therefore, 7 = k^(1/n)
35^6=k, therefore, 35=k^(1/6)
5 x 7 = 35,
therefore, k^(1/m)*k^(1/n)=k^(1/6)
k^[(m+n)/mn]= k^(1/6)
Therefore, (m+n)/mn = 1/6
(Bases are equal, therefore exponents are equal).
Hence, mn/(m+n)=6.

To John E. Franklin:- I shall soon make a list of unsolved problems.

Problem # k + 70

Give the first list of fifteen consecutive non-prime numbers.


Problem # k + 71

Prove that the only set of triplet primes is (3,5,7).
Triplet primes are three consecutive odd numbers which are all prime.
As a rule, (2,3,5) is not included.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#293 2005-12-23 06:15:38

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

Last edited by krassi_holmz (2005-12-23 06:35:26)


IPBLE:  Increasing Performance By Lowering Expectations.

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#294 2005-12-23 06:23:51

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

Here's plot of the function Prime[i+1]-Prime[i] up to 1000.
Actually, it's Prime[Floor[i+1]]-Prime[Floor[i]]


IPBLE:  Increasing Performance By Lowering Expectations.

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#295 2005-12-23 06:29:40

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

Last edited by krassi_holmz (2005-12-23 06:37:12)


IPBLE:  Increasing Performance By Lowering Expectations.

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#296 2005-12-23 06:46:38

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Problems and Solutions

I worked up my own proof, but krassi beat me to it.  I figure I'd post it anyways.

Last edited by Ricky (2005-12-23 07:48:33)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#297 2005-12-23 09:10:59

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

Well done, Ricky!
Your proof is very good.
And I haven't beated you because you was able to give another good proof.


IPBLE:  Increasing Performance By Lowering Expectations.

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#298 2005-12-23 16:08:54

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,425

Re: Problems and Solutions

Well done, krassi_holmz and Ricky!
The solutions to both the problems are correct.
The two proofs are very good too! smile

Problem # k + 72

Player A has one more coin than player B.  Both players throw all of their coins simultaneously and observe the number that come up heads.  Assuming all the coins are fair, what is the probability that A obtains more heads than B?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#299 2005-12-23 16:59:51

Jai Ganesh
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Registered: 2005-06-28
Posts: 48,425

Re: Problems and Solutions

John E. Franklin wrote:

ganesh, do you have a list of the problems
that are unsolved?  That way I can go back
easier and solve some old ones.

The unsolved problems are.....

# k + 1   - Page 5
# k + 28 - Page 8
# k + 31 - Page 8
# k + 38 - Page 9
# k + 40 - Page 9
# k + 42 - Page 9
# k + 47 - Page 10
# k + 48 - Page 10
# k + 59 - Page 11
# k + 72 - Page 13


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#300 2005-12-23 18:05:48

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Problems and Solutions

And for k+47
I think I got it. But I'm not so sure.
The sum of all the ten numbers must be "random" number too, because if remainders by 10 are ramdon we can't say anyting about the remainder of the sum. I think that the probability is 0.1


IPBLE:  Increasing Performance By Lowering Expectations.

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