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Let   x,y,z and  be nonzero real numbers. Find all possible values of

\[x|x|+y|y|+z|z|+xy|xy|+xz|xz|+yz|yz|+xyz|xyz|.\]
Thank you

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 Jun 2, 2024
 #1
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We could use casework for this problem. 

 

First, if x, y, and z are all postive, we have 

x|x|+y|y|+z|z|+xyz|xyz|=xx+yy+zz+xyzxyz=1+1+1+1=4

 

Now, if we have only x is negative, is 0, we have

x|x|+y|y|+z|z|+xyz|xyz|=|x||x|+yy+zz+|x|yz|xyz|=1+1+11=0

 

In the same way, having only y and z being negative will still be 0.

 

If only z is postive, we have

x|x|+y|y|+z|z|+xyz|xyz|=|x||x|+|y||y|+zz+(|x|)(|y|)z|xyz|=11+1+1=0

 

In  the same way, if only x and y are positive, the result will also be 0.

 

If all three x, y, and z are negative, we have

x|x|+y|y|+z|z|+xyz|xyz|=|x||x|+|y||y|+|z||z|+(|x|)(|y|)(|z|)|xyz|=1111=4

 

 

So, the three possible values are 0, -4, and 4. 

 

Thanks! :) 

 Jun 2, 2024

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