3. The area of a quadrilateral is 1. What is the smallest value that the sum of its diagonals can have?
Solution. First, let's find the minimum value for the product of the diagonals of a quadrilateral. For its area, we have the equality: \(2 S = x y \sin \alpha\), where \(x, y\) are the diagonals of the quadrilateral, and \(\alpha\) is the angle between them. Given \(S = 1\), we have \(2 = x y \sin \alpha \leq x y\).

Now, using the inequality of the geometric mean \(2 \sqrt{x y} \leq x + y\), we obtain the minimum value for the sum of the diagonals of the quadrilateral: \(2 \sqrt{2} \leq x + y\).