[ $\left.\begin{array}{c}\text { Equations in integers } \\ \text { [Examples and counterexamples. Constructions }\end{array}\right]$

Do there exist natural numbers $a, b, c, d$ such that $a / b + c / d = 1, a / d + d / b = 2008$?
For example, $a=2009 \cdot(2008 \cdot 2009-1), b=2008 \cdot 2010 \cdot(2008 \cdot 2009-1), c=2008 \cdot 2009-1, d=2008 \cdot 2010$.

## Answer

They exist.