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If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
15 Dec 2009, 14:48

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Difficulty:

95% (hard)

Question Stats:

23% (02:55) correct
77% (04:24) wrong based on 413 sessions

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

Suppose you got the answer of 2 for the values of x and y as 4 and 2.

Great Problem! Since your trying to find the greatest value of X-Y, you just have to assume that Y=0, like Bunel said and then use the "hit and trial" approach like xcusem... Said. The algebratic approach is great too, but I know for me personally it opens up the opportunity for me to make silly mistakes. So I try to not use it unless necessary.

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
08 Feb 2013, 08:27

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axl_oz wrote:

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

Since we need to maximize the value of |x – y|, we can do that in two ways...1)make y negative, which is not possible as per the question...2)make y= 0..putting y=0 you will get an equation in x and on hit and trial method u will get the value of x as 3, which will satisfy the equation.... putting x=3 and y=0, we will get the value of |x – y| as 3.

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
21 Mar 2013, 08:34

Brunel and all,

Is it a rule to apply one value as zero whenever it is given:

1) Both x and y are non-negative integers 2) we need to find the max value of x-y

What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise? _________________

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
21 Mar 2013, 20:56

8

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Expert's post

surya167 wrote:

Brunel and all,

Is it a rule to apply one value as zero whenever it is given:

1) Both x and y are non-negative integers 2) we need to find the max value of x-y

What if we are asked to find the min ? how do we solve those questions and also, what would be the approach for min and max value of x+y ? Can u guys pls advise?

Usually, when you are checking for numbers, you do check for 0. It's often a transition point for patterns. Secondly, the question used the term 'non-negative integers' instead of 'positive integers' - this means 0 would probably have a role to play. There are no such rules but common sense says that we must not ignore 0.

Now, when we look at the equation, 2^x + 2^y = x^2 + y^2, some things come to mind: 1. It is not very easy to find values that satisfy this equation. 2. But there must be some values which satisfy since we are looking for a value of |x – y| 3. If x = y = 2, the equation is satisfied since all terms become equal and |x – y| = 0 which is the minimum value of |x – y|.

Usually, the left hand side will be greater than the right hand side (as discussed in the post, 2^n will usually be greater than x^2 except in very few cases). So we must focus on those 'very few cases'. Also, we need to make x and y unequal.

We know (from the post) that 2^4 = 4^2 is one solution so we could put x = 4 while keeping y = 2. The equation will be satisfied and |x – y| = 2

Now, we also know that 2^x < x^2 when x = 3. So is there a solution there as well? The difference between 2^3 and 3^2 is of 1 so can we create a difference of 1 between the other two terms? Sure! If y = 0, then 2^0 = 1 but 0^2 = 0. So another solution is 2^3 + 2^0 = 3^2 + 0^2. Here, |x – y| = 3 which is the maximum difference.

The reason we can be sure that there are no other values is that as you go ahead of 4 on the number line, 2^n will be greater than n^2 (again, discussed in the post). So both left hand side terms will be greater than the right hand side terms i.e. 2^x > x^2 and 2^y > y^2. So, for no other values can we satisfy this equation.

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
05 Dec 2013, 23:37

axl_oz wrote:

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

my answer: |x – y| for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 right D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
06 Dec 2013, 01:52

Expert's post

misanguyen2010 wrote:

axl_oz wrote:

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

my answer: |x – y| for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 right D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

My answer is C.

Please note that the correct answer is D, not C. _________________

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
06 Dec 2013, 09:47

Bunuel wrote:

misanguyen2010 wrote:

axl_oz wrote:

If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative integers, what is the greatest possible value of |x – y|?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

This is a challenge problem on MGMAT. I do not have the answer to the question... But on solving the problem I got the answer of 2 (not sure if it is right)

my answer: |x – y| for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 right D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

My answer is C.

Please note that the correct answer is D, not C.

Hi thank you for your reply. I explained what i confused. Of course I read previous answers and all chose D. However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer. Please help!

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
07 Dec 2013, 04:52

Expert's post

misanguyen2010 wrote:

Bunuel wrote:

misanguyen2010 wrote:

my answer: |x – y| for x and y 2^x + 2^y = x^2 + y^2 A. x= y x=y=1 2 + 2 = 1 + 1 wrong B. 1 x=2, y=1 4 + 2 = 4 + 1 wrong C. 2 x=3, y=1 8 + 2 = 9 + 1 right D. 3 x=4, y =1 16 + 2 = 16 + 1 wrong E. 4 x=5, y =1 32 + 2 = 25 + 1 wrong

My answer is C.

Please note that the correct answer is D, not C.

Hi thank you for your reply. I explained what i confused. Of course I read previous answers and all chose D. However, from what i found, i chose C. That s why i posted here. I dont know which is wrong in my answer. Please help!

To get the greatest value of |x-y| as 3 consider x=3 and y=0. Notice that these values satisfy 2^x + 2^y = x^2 + y^2 --> 2^3 + 2^0 =9= 3^2 + 0^2.

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte [#permalink]
28 Dec 2013, 03:25

Here is how I done it:

1) If |x-y| needs to be max then Y=0, because Y² is only positive 2) Check the answers, those are only integers, you are therefore looking for an integer 3) You have the equation 2^x +1 = X² 4) Use the different choices and you will see that only 3 matches.

Answer D

Hope it helps! _________________

Think outside the box

gmatclubot

Re: If 2^x + 2^y = x^2 + y^2, where x and y are nonnegative inte
[#permalink]
28 Dec 2013, 03:25

I attended a portfolio workshop hosted by Business Design club today. Competing against thousands of MBA students with the entire world, you need more than your resume and coverletter...

so actually alongside the MBA studies, I am studying for personal trainer exam in December as a side. I can basically only read when I’m in the subway...