6.92*. For which $n$ does there exist a convex $n$-gon, one of whose sides has length 1, and the lengths of all diagonals are integers?
6.92. We will prove that $n \leqslant 5$. Let $AB=1$, and let $C$ be a vertex not adjacent to either $A$ or $B$. Then $|AC - BC| < AB = 1$. Therefore, $AC = BC$, i.e., point $C$ lies on the perpendicular bisector of side $AB$. Thus, besides vertices $A, B, C$, the polygon can have only two more vertices.

An example of a pentagon with the required property is shown in Fig. 6.19. Let us explain how it is constructed. $ACDE$ is a rectangle, $AC = ED = 1$, and $\angle CAD = 60^{\circ}$. Point $B$ is defined by the condition $BE = BD = 3$.

An example of a quadrilateral with the required property is the rectangle $ACDE$ in the same figure.