1. Find the maximum possible area of a quadrilateral in which the product of any two adjacent sides is 1.
1. Find the maximum possible area of a quadrilateral for which the product of any two adjacent sides is 1.

OTBET: 1.

SOLUTION.

Let the quadrilateral have sides $a, b, c, d$. Then $a b=b c=c d=d a=1$. From the equality $a b=b c$, it follows that $a=c$, and from the equality $b c=c d$, we get that $b=d$. Therefore, the given quadrilateral is a parallelogram. Let $\alpha$ be the angle between sides $a$ and $b$. Then $S=a \cdot b \sin \alpha$, and the area is maximized if $\sin \alpha=1$. Thus, $S=a \cdot b=1$.