14.2. (Jury, Belgium, 82). Among all triangles of a given perimeter, find the one for which the radius of the inscribed circle is maximized.
14.2. Let $a, b, c$ be the sides of a triangle with a given semiperimeter $p$, $S$ be its area, and $r$ be the radius of the inscribed circle. Then, by the theorem of means, we have

$$
\begin{aligned}
(r p)^{2}=S^{2}=p(p-a)(p-b)(p-c) & \leqslant \\
& \leqslant p\left(\frac{(p-a)+(p-b)+(p-c)}{3}\right)^{3}=\frac{p^{4}}{27}
\end{aligned}
$$

from which $r \leqslant p / \sqrt{27}$, and the greatest value of $r$ is attained when $p-a=p-b=p-c$, i.e., when the triangle is equilateral.