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**mathsyperson****Moderator**- Registered: 2005-06-22
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Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Administrator**- Registered: 2005-06-28
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mathsyperson is correct

Problem # k + 78

A, B, and C are partners in a business. Their shares of capital are 1/2:1/3:1/4. A withdraws half of his capital after 15 months. After 15 months more,the profit of $ 4340 is divided between them propotional to their capital. What is C's share of the profit?

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
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Problem # k + 79

Show that the set (1, 11, 101, 1001,......10^n+1) where n≥10 has more non-prime numbers than primes.

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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**krassi_holmz****Real Member**- Registered: 2005-12-02
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*Last edited by krassi_holmz (2006-01-04 23:30:18)*

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Administrator**- Registered: 2005-06-28
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Outstanding! krassi_holmz, you deserve to be complimented for that!

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: 23,219

Probelm # k + 80

A line 2x + 3y + 1 = 0 tocuhes a circle C at (1, -1). Another circle cuts circle C orthogonally and the end points of its diameter are (0, -1), (-2, 3). Find the equation of the circle C.

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

Thank you. For the next problem I'll leave it to somebody else.

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

Interestingly, none has replied to Problem # k + 78, which I thought was relatively simple

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

krassi_holmz wrote:

Thank you. For the next problem I'll leave it to somebody else.

Me too. I've answered far too many of these, so let's give someone else a turn.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

Probelm # k + 81

If [{(sinθ)/(1+cosθ)} + {(1+cosθ)/(sinθ)}]= 4, what is the value of θ if

0 degrees≤θ≤90 degrees?

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
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I'm not sure.

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

krassi_holmz, you are correct! Why are you uncertain about a correct answer? Well done, krassi_holmz.

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

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**ganesh****Administrator**- Registered: 2005-06-28
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Problem # k + 82

If x^4 + y^4 = 17 and x + y =1, what is the value of x²y²-2xy?

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

x^2+y^2=x^2+y^2+2xy-2xy=(x+y)^2-2xy=1-2xy.

x^4+y^4=17=x^2^2+y^2^2+2x^2y^2-2x^2y^2=(x^2+y^2)^2-2x^2y^2=(1-2xy)^2-2x^2y^2 = 1-4xy+4x^2y^2-2x^2y^2=1-4xy+2x^2y^2 =>

2x^2y^2-4xy=17-1=16

x^2y^2-2xy=16/2=8

IPBLE: Increasing Performance By Lowering Expectations.

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

I might be missing something obvious, but...

1/2 + 1/3 + 1/4 = 13/12? Doesn't one of the partners in k+78 own a 1/12 imaginary share?

I think that this problem could only be solved if we knew who was holding the fake share.

*Last edited by irspow (2006-01-08 04:54:55)*

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

krassi_holmz is right!

To irspow : In problem # k + 78, when it is stated that the shares are in the ration a:b:c, it need not be true that a+b+c be equal to 1. a's share of the total would be a/(a+b+c), b's share would be b/(a+b+c) and c's c/(a+b+c).

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

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**ganesh****Administrator**- Registered: 2005-06-28
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Problem # k + 83

Four horses are tethered at 4 corners of a square field of side 70 metres so that they just cannot reach one another. What is the area left ungrazed by the horses?

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

70^2 - 45^2 Pi ?

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

krassi_holmz, you are given another chance.

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

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

*Last edited by Ricky (2006-01-09 03:40:26)*

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

*Last edited by irspow (2006-01-09 13:39:29)*

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**ganesh****Administrator**- Registered: 2005-06-28
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Well done, irspow

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: 23,219

Problem # k + 84

Two spheres of radii 6 cm and 1 cm are inscribed in a right circular cone. The bigger sphere touches the smaller sphere and also the base of the cone. What is the height of the cone?

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

I'll take an ill-attempted stab.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 23,219

irspow is correct! Although the actual solution is arrived at differently!

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

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