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# Inequalities

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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