[ Convex Polygons ]

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?

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Let's prove that $n \leq 5$. Suppose $AB=1$, and $C$ is 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 the figure. Let's 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.

![](https://cdn.mathpix.com/cropped/2024_05_06_18f5fa9123f1d66ab2f1g-30.jpg?height=1100&width=580&top_left_y=-1&top_left_x=745)