The total number of sides of two polygons is 24, and the total number of diagonals is 109. How many sides and diagonals does each polygon have?
If one polygon has $n$ sides, then the other is a 24 - $n$ sided polygon, in the first the number of diagonals is $\frac{n(n-3)}{2}$, in the second $\frac{(24-n)(21-n)}{2}$, so

$$
\frac{n(n-3)}{2}+\frac{(24-n)(21-n)}{2}=109
$$

or

$$
n^{2}-24 n+143=0
$$

from which

$$
n_{1}=13 \quad \text { and } \quad n_{2}=11 \text {. }
$$

One polygon has 13 sides, the other has 11, the former has 65 diagonals, the latter has 44.

Number of solutions: 39.

(Vilmos Erdös, Budapest.)