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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: 1,055

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)*

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

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).

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

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?

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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**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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

krassi_holmz, you are given another chance.

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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**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: 1,055

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

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

Well done, irspow

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

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: 1,055

I'll take an ill-attempted stab.

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

Probelm # k + 85

If u and v are the roots of the equation x² + ax + b = 0, what are roots of the equation x² -ax + b = 0?

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

That's an interesting one. I think it's something like this:

Why did the vector cross the road?

It wanted to be normal.

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

I'm not sure, this is what I get:

*Last edited by Ricky (2006-01-11 04:16:58)*

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

I think our answers are the same, and I've just taken a long way round.

Why did the vector cross the road?

It wanted to be normal.

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

Yea, I guess they are. They looked completely different at first glance.

"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: 1,055

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 1,055

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 1,055

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 1,055

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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**irspow****Member**- Registered: 2005-11-24
- Posts: 1,055

darn that k+42, I just cant figure it. Please someone, put me out of my misery. I think that you have to incorporate a geometric series somehow, but everything that I try turns to nonsense.

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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

Yes, k+42 is an incredibly difficult one. I think it could probably be solved brutally by getting excel to do all the calculations for you, but it's still tough.

Also, I think your answer to k+40 is wrong. I remember it being *much* smaller.

Why did the vector cross the road?

It wanted to be normal.

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**ryos****Member**- Registered: 2005-08-04
- Posts: 394

I didn't check if this one has already been solved, but since irspow asked about it, I gave it a go.

*Last edited by ryos (2006-01-14 16:15:25)*

El que pega primero pega dos veces.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

Four days Pongal break, and so many solutions posted! I shall reply to all of them after I return from Holiday on Jan 16. mathsyperson is right, irspow's solution to problem # k + 40 isn't correct. It *is much smaller. *. Same about ryos' solution to problem # k + 42.

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

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