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

\[
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3 \text{ ? }
\]
First, assume that the length of the third side c is not subject to any conditions. The area of a triangle with sides a and b and the angle $\alpha$ between them is determined by the formula $1 / 2$ ab sin $\alpha$. If a and b are subject to the conditions 0 , then the product ab $\sin \alpha$ will be maximal when each of the factors is maximal, i.e., when $a=1, b=2, \sin \alpha=1\left(\alpha=90^{\circ}\right)$. Thus, the triangle with the largest area is a right triangle with legs 1 and 2. The third side of this triangle $c=\sqrt{5}$ satisfies the conditions 2 with 3 . Therefore, among all triangles whose sides are subject to the condition $0<a<=1<=b<=2<=c<=3$, the one with the maximum area will be the same right triangle. This area is equal to 1.

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