1. What is the maximum number of sides a convex polygon can have if all its diagonals are of equal length?
1. The answer is 5. First, note that a regular pentagon is an example of such a polygon. Now let a convex $n$-gon $A_{1} A_{2} \ldots A_{n}$ be given for $n \geqslant 6$. The quadrilateral $A_{1} A_{2} A_{n-2} A_{n-1}$ is also convex; from the triangle inequality we have $A_{n-1} A_{2} + A_{1} A_{n-2} > A_{n-1} A_{1} + A_{n-2} A_{2}$, so in the considered $n$-gon, not all diagonals $A_{n-1} A_{2}, A_{1} A_{n-2}$, $A_{n-1} A_{1}$, and $A_{n-2} A_{2}$ can be of equal length.