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**rimi****Member**- Registered: 2006-02-21
- Posts: 17

For problem#K+101

It does not hold for n=2... as (2!+1)=3,which is a prime number.

Ganesh ,by infinitely many values of n , do you mean large values of n? Then what is the lower limit for n?

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

rimi, your solution to Problem # k + 100 is correct

In Problem # 101, infintely many values means an unending sequence. Like the list of prime numbers, powers of the number 2 etc. For both n=1, n=2, and n=3, the resultants are prime numbers. However, for n=4, n=5, n=6, n=7 etc., the value of n! + 1 is composite.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 102

Three-fourth of a number is equal to 60% of another

number, and the difference between the two numbers is 20. What

is the sum of the two numbers?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

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

By taking the route of a - b = 20, instead of b - a = 20, you can also get an answer of -180.

Why did the vector cross the road?

It wanted to be normal.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

Nice catch mathsyperson, can I still be half right? Maybe it should have read, "What are the sums of the numbers?". Hey, did I just get tricked!?

*Last edited by irspow (2006-02-23 10:20:06)*

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

You can be almost fully right, really. You did all the working, and I just stuck a little comment on the end. And I think the reason why ganesh didn't phrase it like that was because he missed it himself.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

mathsyperson is right! Although I explore the possibilites of other solutions, this time I didn't think of that

Well done, irspow and mathsyperson

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 103

A circle is inscribed within an equilateral triangle and another is circumscribed. Calculate the ratio of the area of the incircle to the circumcircle.

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 104

What is the maximum slope of the curve y = -x³ - 3x² + 9x - 27 and what point is it?

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

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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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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Well done, mathsyperson

Problem # k + 105

A group of bess equal in number to the square root of half the whole swarm alighted on a jasmine bush, leaving behind 8/9 of the swarm. And only one bee circled a lotus for it was attracted by the buzzing of a sister bee that was so careless as to fall into the trap of the fragrant flower. How many bees were there in the swarm?

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 106

Four brother have 45 Dollars. If the money of the first is inreased by 2 Dollars and the money of the second is decreased by 2 Dollars, and the money of the third is doubled and the money of the fourth is halved, then all of them will have the same amount of money. How much does each have?

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

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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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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 107

The price of a water-melon is 50 cents, an apple is 10 cents and a plum is 1 cent. Five dollars were used to buy 100 items of different kinds of fruit. How many pieces of each type were bought?

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 108

What is the sum of the first 50 terms common to the series 15,19,23 ... and 14,19,24 ... ?

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

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

Possibly a bit unconventional...

But nothing unconventional here.

Why did the vector cross the road?

It wanted to be normal.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,691

(19 pages, wowee! Maybe you could start a new topic?)

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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

I was actually wondering that for quite a while, MathIsFun. Wouldn't it be more organized if Ganesh did one question per topic? I mean, we got an entire section of the forum for it, why not?

"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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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

To mathsyperson :- Funnily, your solution to Problem # k + 107 is correct, although unconventional, as you put it. Please read the Problem # k + 108 again before posting your solution.

To MathsIsFun :- Good suggestion, worth considering.

To Ricky :- One question per topic is fine. But what do we do with the problems already posted? Put them all in an 'Assorted' or 'Miscellaneous' topic? Good suggestion, worth considering.

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

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

Do you mean this is the end of "Problems and solutions"?

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

I hope not. Some problems would be posted here in the future too. There are some unanswered problems for which solutions would have to be posted. Hence, this is not the end of 'Problems and solutions'. This topic shall remain the precursor of other topics.

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

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,867

Problem # k + 109

Prove that every number of the form a[sup]4[/sup]+4 is a composite number (a≠1).

(This problem was posed by the eminent French mathematician Sophie Germain).

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

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

It's fairly simple apart from when a ends in 5.

When mod(10) a = 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, mod(10)(a[sup]4[/sup] + 4) = 4, 5, 0, 5, 0, 9, 0, 5, 0 and 5 respectively.

Numbers that end in 4, 5 or 0 are never prime (apart from 5) so that proves it for all values of a except for ##5. But proving it for that is quite difficult.

Why did the vector cross the road?

It wanted to be normal.

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