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Is x^4 + y^4 > z^4? 1. x^2 + y^2 > z^2 2. x + y > z

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Is x^4 + y^4 > z^4? 1. x^2 + y^2 > z^2 2. x + y > z [#permalink] New post 05 Aug 2005, 23:26
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Is x^4 + y^4 > z^4?

1. x^2 + y^2 > z^2
2. x + y > z
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Re: DS-integers,,,,, [#permalink] New post 06 Aug 2005, 05:05
(A) for me.

In statement 2, x or y may be negative. Consider x,y,z=-10,-9,-20 and x,y,x=2,1,1. You will get different results.

In (1), the value of x^2, y^2 and z^2 must be +ve and if the sum of x^2 and y^2 is greater than z^2, then we can conclude that the same will hold true for the stem.
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 [#permalink] New post 06 Aug 2005, 13:18
But (x^2 + y^2)^2 < (x^4 + y^4)

you have forgotten the cross terms !
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 [#permalink] New post 07 Aug 2005, 13:03
Don't forget the cross terms when you square the left hand side !

Consider x=4, y=5, z=6

x + y > z
x^2 + y^2 > z^2
but
x^4 + y^4 < z^4

Consider x= 4, y=5 , z=1
x + y > z
x^2 + y^2 > z^2
and
x^4 + y^4 > z^4

Neither statement on its own or both is sufficient.
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 [#permalink] New post 07 Aug 2005, 23:47
OA is E.

I was wondering if there is a way other than plugging in numbers.
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 [#permalink] New post 08 Aug 2005, 09:35
Just look at all the cross-terms in the algebraic expansion.
Put in = instead of > to make it easier.

x + y = z
(x+y)^2 = z^2
x^2 + y^2 = z^2 - 2xy
(x^2 + y^2)^2 = (z^2 - 2xy)^2
x^4 + y^4 = z^4 + 2x^2y^2 - 4xyz^2
x^4 + y^4 = z^4 + 2x^2y^2 - 4xy(x^2+y^2)
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 [#permalink] New post 08 Aug 2005, 21:10
richardj wrote:
Just look at all the cross-terms in the algebraic expansion.
Put in = instead of > to make it easier.

x + y = z
(x+y)^2 = z^2
x^2 + y^2 = z^2 - 2xy
(x^2 + y^2)^2 = (z^2 - 2xy)^2
x^4 + y^4 = z^4 + 2x^2y^2 - 4xyz^2
x^4 + y^4 = z^4 + 2x^2y^2 - 4xy(x^2+y^2)


from that point on, how do we judge insuff from both sides (less than/greater than)? do we have to try different values?
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 [#permalink] New post 08 Aug 2005, 21:43
Hmm, good point.

Not sure I can actually finish the proof !

And you would never write all that out in the test itself.

The point is that even with x,y>0, there are terms with positive and negative signs involving powers of x and y on the right. So that makes it quite likely there are two different cases.
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Re: DS-integers,,,,, [#permalink] New post 09 Aug 2005, 20:54
1. x^2 + y^2 > z^2
(x^2+y^2)^2>z^4
x^4+y^4+2x^2y^2>z^4
x^4+y^4>z^4-2x^2y^2
Since x^2y^2>0, we don't know if x^4+y^4>z^4.
Insufficient

2. x + y > z
They could be different signs. Say both x and y are small positive numbers and z is a large negative. Then x^4+y^4 could be smaller than z^4. But the opposite is also possible. So insufficient.

Combined:
Not very easy to prove. The best way to go, in my opinion is by counter examples. As long as you can find a set of numbers where both 1 and 2 are satisfied to answer No to the stem then you are all set. Because it is easy to find examples to say Yes to the stem, finding the No answer would mean the answer is not unique and that 1 and 2 combined are not sufficient.
Anyways you could try with the closest numbers such as 3, 3, 4, where 3+3>4, 3^2+3^2>4^2, but 3^4+3^4<4^4.

Therefore the answer is E.
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Re: DS-integers,,,,,   [#permalink] 09 Aug 2005, 20:54
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