Problems and Solutions

rimi · 2006-02-22 19:26:34

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?

ganesh · 2006-02-22 20:03:10

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.

ganesh · 2006-02-22 20:41:05

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?

irspow · 2006-02-23 09:11:14

mathsyperson · 2006-02-23 09:55:24

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

irspow · 2006-02-23 10:11:01

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)

mathsyperson · 2006-02-23 10:25:38

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.

ganesh · 2006-02-23 18:11:12

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

ganesh · 2006-02-23 18:43:12

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.

ganesh · 2006-02-23 21:50:00

Problem # k + 104

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

mathsyperson · 2006-02-24 05:12:42

ganesh · 2006-02-24 23:56:40

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?

ganesh · 2006-02-25 00:40:29

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?

mathsyperson · 2006-02-25 09:17:57

ganesh · 2006-02-25 17:31:21

ganesh · 2006-02-25 17:38:44

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?

ganesh · 2006-02-26 00:13:53

Problem # k + 108

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

mathsyperson · 2006-02-26 06:48:06

Possibly a bit unconventional...

But nothing unconventional here.

MathsIsFun · 2006-02-26 08:38:42

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

Ricky · 2006-02-26 15:56:27

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?

ganesh · 2006-02-26 16:49:34

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.

krassi_holmz · 2006-02-26 22:18:34

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

ganesh · 2006-02-26 22:26:23

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.

ganesh · 2006-02-28 16:28:18

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).

mathsyperson · 2006-03-01 03:34:28

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.

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