23. What is the maximum area that a triangle with sides \(a, b, c\) can have, given the following constraints:

\[
0 \leqslant a \leqslant 1 \leqslant b \leqslant 2 \leqslant c \leqslant 3 \text { ? }
\]
23. Answer: 1. Among triangles with two sides $a, b$ that satisfy the conditions $0<a \leqslant 1,1 \leqslant b \leqslant 2$, the one with the largest area is the right triangle with legs $a=1, b=2$ (indeed, $s \leqslant a b / 2 \leqslant 1$, since the height dropped to side $b$ is no more than $a$). The third side of this triangle $c=\sqrt{5}$ satisfies the condition $2 \leqslant c \leqslant 3$, therefore, among all the considered triangles, it has the largest area.