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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****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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?

Character is who you are when no one is looking.

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**ganesh****Moderator**- 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.

Character is who you are when no one is looking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

*Last edited by krassi_holmz (2006-01-04 23:30:18)*

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

Outstanding! krassi_holmz, you deserve to be complimented for that!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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.

Character is who you are when no one is looking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

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

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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

Character is who you are when no one is looking.

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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****Moderator**- Registered: 2005-06-28
- Posts: 13,224

Probelm # k + 81

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

0 degrees≤θ≤90 degrees?

Character is who you are when no one is looking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

I'm not sure.

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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

Character is who you are when no one is looking.

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**ganesh****Moderator**- 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?

Character is who you are when no one is looking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

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: 455

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****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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).

Character is who you are when no one is looking.

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**ganesh****Moderator**- 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?

Character is who you are when no one is looking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

70^2 - 45^2 Pi ?

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

krassi_holmz, you are given another chance.

Character is who you are when no one is looking.

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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: 455

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

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

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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?

Character is who you are when no one is looking.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 455

I'll take an ill-attempted stab.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,224

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

Character is who you are when no one is looking.

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