13.229. Find four numbers forming a proportion, given that the sum of the extreme terms is 14, the sum of the middle terms is 11, and the sum of the squares of these four numbers is 221.
Solution.

Let the proportion be $\frac{a}{b}=\frac{c}{d}$. Given $a+d=14$, $c+b=11, a^{2}+b^{2}+c^{2}+d^{2}=221$.

Solving the system $\left\{\begin{array}{l}a d=b c, \\ a+d=14, \\ b+c=11, \\ a^{2}+b^{2}+c^{2}+d^{2}=221,\end{array}\right.$

we find $a=12 ; b=8 ; c=3 ; d=2$.

Answer: $12,8,3,2$.