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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

We can rearrange the equation, putting all the \(x\)’s on one side and all the \(y\)’s on the other side: \(2^x - x^2 = y^2 - 2^y\)

Now, list the values of \(2^n\) and \(n^2\) for the first several nonnegative integers \(n\). In fact, go ahead and compute the differences both ways (both \(2^n - n^2\) and \(n^2 - 2^n\)). \(n\) \(2^n\) \(n^2\) \(2^n - n^2\) \(n^2 - 2^n\) 0 1 0 1 -1 1 2 1 1 -1 2 4 4 0 0 3 8 9 -1 1 4 16 16 0 0 5 32 25 7 -7 6 64 36 28 -28 From this point on, \(2^n\) grows much faster than \(n^2\), so the differences explode. This means that in order to have a valid equation \((2^x - x^2 = y^2 - 2^y)\), we will have to use small values of the integers. We want values in the \(2^n - n^2\) column to match values in the \(n^2 - 2^n\) column, and to maximize the value of \(|x - y|\), we want to pick values from different rows - as far apart as possible.

If we pick \(x = 0\) and \(y = 3\) (or vice versa), then we get a valid equation: \(2^0 - 0^2 = 3^2 - 2^3\) \(1 - 0 = 9 - 8\)

These values of \(x\) and \(y\) are as far apart as possible, so we get \(|x - y| = 3\).

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

We can rearrange the equation, putting all the \(x\)’s on one side and all the \(y\)’s on the other side: \(2^x - x^2 = y^2 - 2^y\)

Now, list the values of \(2^n\) and \(n^2\) for the first several nonnegative integers \(n\). In fact, go ahead and compute the differences both ways (both \(2^n - n^2\) and \(n^2 - 2^n\)). \(n\) \(2^n\) \(n^2\) \(2^n - n^2\) \(n^2 - 2^n\) 0 1 0 1 -1 1 2 1 1 -1 2 4 4 0 0 3 8 9 -1 1 4 16 16 0 0 5 32 25 7 -7 6 64 36 28 -28 From this point on, \(2^n\) grows much faster than \(n^2\), so the differences explode. This means that in order to have a valid equation \((2^x - x^2 = y^2 - 2^y)\), we will have to use small values of the integers. We want values in the \(2^n - n^2\) column to match values in the \(n^2 - 2^n\) column, and to maximize the value of \(|x - y|\), we want to pick values from different rows - as far apart as possible.

If we pick \(x = 0\) and \(y = 3\) (or vice versa), then we get a valid equation: \(2^0 - 0^2 = 3^2 - 2^3\) \(1 - 0 = 9 - 8\)

These values of \(x\) and \(y\) are as far apart as possible, so we get \(|x - y| = 3\).

Answer: D

Hello there Bunuel.

I'm new here!

Do we have any other method other than this one to solve this question?

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

We can rearrange the equation, putting all the \(x\)’s on one side and all the \(y\)’s on the other side: \(2^x - x^2 = y^2 - 2^y\)

Now, list the values of \(2^n\) and \(n^2\) for the first several nonnegative integers \(n\). In fact, go ahead and compute the differences both ways (both \(2^n - n^2\) and \(n^2 - 2^n\)). \(n\) \(2^n\) \(n^2\) \(2^n - n^2\) \(n^2 - 2^n\) 0 1 0 1 -1 1 2 1 1 -1 2 4 4 0 0 3 8 9 -1 1 4 16 16 0 0 5 32 25 7 -7 6 64 36 28 -28 From this point on, \(2^n\) grows much faster than \(n^2\), so the differences explode. This means that in order to have a valid equation \((2^x - x^2 = y^2 - 2^y)\), we will have to use small values of the integers. We want values in the \(2^n - n^2\) column to match values in the \(n^2 - 2^n\) column, and to maximize the value of \(|x - y|\), we want to pick values from different rows - as far apart as possible.

If we pick \(x = 0\) and \(y = 3\) (or vice versa), then we get a valid equation: \(2^0 - 0^2 = 3^2 - 2^3\) \(1 - 0 = 9 - 8\)

These values of \(x\) and \(y\) are as far apart as possible, so we get \(|x - y| = 3\).

Answer: D

Hello there Bunuel.

I'm new here!

Do we have any other method other than this one to solve this question?

I don't agree with the explanation. Take X=3 Y =1 X-Y = 2 , equality holds true. Take X=4 Y=1 X-Y =3 , equality doesn't hold true..

True answer should be C not D.. Am i missing something ?

Please re-read the solution. Where does it say that the greatest possible value of |x-y| is for x=4 and y=1? It's for x=0 and y=3 (or vice versa).
_________________

X and Y are nonnegative integers and we need values of X and Y such that mod (X-Y) is maximum. In order to get maximum difference, one value must be zero, so we can assume Y = 0. Then,looking at all options only X=3 satisfies the equation.

X and Y are nonnegative integers and we need values of X and Y such that mod (X-Y) is maximum. In order to get maximum difference, one value must be zero, so we can assume Y = 0. Then,looking at all options only X=3 satisfies the equation.

So, if x=0,y=4 then equation becomes 2^0 + 2^4 = 0^2 + 4^2 (Each side equals 16) now,the difference between x and y becomes 4. So,should the answer not be 4?

So, if x=0,y=4 then equation becomes 2^0 + 2^4 = 0^2 + 4^2 (Each side equals 16) now,the difference between x and y becomes 4. So,should the answer not be 4?