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Let x and y be positive real numbers. Find the minimum value of

 

\(\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 2018 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 2018 \right).\)

 

I really do not know where to start without involving huge expressions.  

 

Naturally, however, if you could write \(y+\frac{1}{x}\) in terms of \(x+\frac{1}{y}\), or vice versa, that would really help.  

 

Thanks for any replies!
 

 Jun 20, 2021
 #1
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The minimum value is -4036.

 Jul 11, 2021

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