Problem - 3129
Use the Arithmetic Mean-Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.
Let $x$ be the lengths of the corner being cut.
\begin{align}
V &= x(25-2x)(15-2x)\\
&= \frac{1}{4}(4x)(25-2x)(15-2x)\\
&\le \frac{1}{4}\cdot\Big(\frac{4x+25-2x+15-2x}{3}\Big)^3\\
&= \boxed{\frac{1}{4}\cdot\Big(\frac{50}{3}\Big)^3}
\end{align}