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# Is x^4 + y^4 ?z^4? 1) x^2+y^2 2) x+y >z

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Is x^4 + y^4 ?z^4? 1) x^2+y^2 2) x+y >z [#permalink]  15 Jul 2008, 08:40
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Is x^4 + y^4 ?z^4?
1) x^2+y^2
2) x+y >z
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 06:08
MamtaKrishna, there seems to be some type here.
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 07:05
Sorry there !!!

Is x^4 + y^4 >z^4?
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 07:20
Is there a typo in statement 1?
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 09:03
Ooops

Yeah the Q is

Is x^4 + y^4 ?z^4?
1) x^2+y^2 > z^2
2) x+y >z
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 15:20

I will use the substitution method here.

From Stmt1: x^2 + y^2 > z^2. Hence, for any pair of values of x, y and z, x^4 + y^4 will be greater than z^4.

Stmt2: The same logic as for stmt 1.
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Re: Is x^4 + y^4 ?z^4? [#permalink]  16 Jul 2008, 16:27
scthakur wrote:

I will use the substitution method here.

From Stmt1: x^2 + y^2 > z^2. Hence, for any pair of values of x, y and z, x^4 + y^4 will be greater than z^4.

Stmt2: The same logic as for stmt 1.

St 1 Explanation is OK!

But in St 2, missed some numbers. ie:

Positive Numbers:
2+2>3 , but 2^4+2^4<3^4

And Negative Numbers.
(-1)+(-1)>-3, but (-1)^4+(-1)^4<-3^4

IMO A
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Re: Is x^4 + y^4 ?z^4? [#permalink]  17 Jul 2008, 04:05
Even i picked A.
But OA given is E.
It is a Q from Gmat prep test 2 ..
But i dont understnd how A is not sufficient..
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Re: Is x^4 + y^4 ?z^4? [#permalink]  17 Jul 2008, 05:12
Statement 1 :
x^2 + Y^2 > Z^2

squaring both sides

x^4 + 2x^2 y^2 + y^2 ( exp of (a+b)^2)

=> x^4 + y^2 > z^4 - 2x^2 y^2

but we dont know anythign bout teh term -2x^2 y^2

Therefore, unless - 2x^2 y^2 =0, we cant be sure
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Re: Is x^4 + y^4 ?z^4? [#permalink]  17 Jul 2008, 08:07
MamtaKrishnia wrote:
Even i picked A.
But OA given is E.
It is a Q from Gmat prep test 2 ..
But i dont understnd how A is not sufficient..

We all know that whole numbers will satisfy both statements and the question stem but try these: z = 1/3 x = 1/4 and y = 1/4
(1) z^2 < x^2 + y^2 --> 1/9 < 1/16 + 1/16 --> 1/9 < 1/8 satisfies statement 1
but when entering this into the above question, you get:
1/81 > 1/128 INSUFF

(2) the same numbers can be used for statement 2
INSUFF

Together: same numbers so therefore INSUFF --> E
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Re: Is x^4 + y^4 ?z^4? [#permalink]  19 Jul 2008, 06:07
brokerbevo wrote:
We all know that whole numbers will satisfy both statements and the question stem but try these: z = 1/3 x = 1/4 and y = 1/4
(1) z^2 < x^2 + y^2 --> 1/9 < 1/16 + 1/16 --> 1/9 < 1/8 satisfies statement 1
but when entering this into the above question, you get:
1/81 > 1/128 INSUFF

(2) the same numbers can be used for statement 2
INSUFF

Together: same numbers so therefore INSUFF --> E

hmmm!!!
I just couldn't come up with such a set...
thanks guys ..
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Re: Is x^4 + y^4 ?z^4? [#permalink]  19 Jul 2008, 09:25
MamtaKrishnia wrote:
brokerbevo wrote:
We all know that whole numbers will satisfy both statements and the question stem but try these: z = 1/3 x = 1/4 and y = 1/4
(1) z^2 < x^2 + y^2 --> 1/9 < 1/16 + 1/16 --> 1/9 < 1/8 satisfies statement 1
but when entering this into the above question, you get:
1/81 > 1/128 INSUFF

(2) the same numbers can be used for statement 2
INSUFF

Together: same numbers so therefore INSUFF --> E

hmmm!!!
I just couldn't come up with such a set...
thanks guys ..

another set was 10+11 > 17

this was not too hard to calculate...
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Re: Is x^4 + y^4 ?z^4? [#permalink]  19 Jul 2008, 10:34
brokerbevo wrote:
We all know that whole numbers will satisfy both statements and the question stem

I guess it's already been pointed out, but that's not always the case. Try x = 6, y = 8, z = 9.

It seems easiest to me to notice the relationship between the statements and triangles. The sum of two sides of a triangle is always greater than the third side, but the sum of the squares of the sides is not always greater than the square of the third side- the Pythagorean Theorem tells us that the sum of the squares of the two shortest sides is never greater than the square of the longest side if we look at the sides of a right angled triangle. That is, we've all seen dozens of examples (right angled triangles) where a+b > c but a^2 + b^2 = c^2.

So, let x^2, y^2 and z^2 be the sides of a 3-4-5 triangle and you get an example immediately that proves the answer to the original question should be E:

x^2 = 3 (i.e. x = root(3))
y^2 = 4 (i.e. y = 2)
z^2 = 5 (i.e. z = root(5))

It's easy to check that x+y > z, and we already know that x^2+y^2 > z^2 and x^4 + y^4 = z^4 (i.e. it's not greater than z^4). E.
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Re: Is x^4 + y^4 ?z^4?   [#permalink] 19 Jul 2008, 10:34
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