131. How many sides in a convex polygon can be equal in length to the longest diagonal?
131. Answer: no more than two. An example of a polygon with two sides equal to the largest diagonal is shown in Fig. 61.

Suppose there are more than two such sides. We select two of them, $A B$ and $C D$, which do not share any vertices (this is possible, [^3] since a polygon with diagonals is not a triangle). Then at least one of the diagonals $A C$ or $B D$ is longer than side $A B$: if these diagonals intersect at some point $K$, then

$$
\begin{aligned}
& A C+B D=A K+K C+B K+ \\
& \quad+K D>A B+C D=2 A B
\end{aligned}
$$

Fig. 61