Recently, I have found these two problems:
The $12$ edges of a parallelepiped are integers. Given that their sum is $100$, what is the maximum value of the volume, or: $$V=abc$$
I have posted both hare because I think the methods for finding the solutions are similar (let me know if I have to open another question).
I can't completely figure out a solution to the first problem, while for the second, I can say: $a+b+c=25 \Longleftrightarrow c=25-a-b$ and so, substituing: $$V=abc=ab\cdot(25-a-b)=25ab-a^2b-ab^2$$ But here I am another time stucked. Any iodea? Do I have to use inequalities such as C-S and Am-GM?
Thank you.
AM-GM inequality can be used here.
We have $$\frac{a+b+c}{3} \ge \sqrt[3]{abc}$$
Hence $$\left(\frac{a+b+c}{3} \right)^3 \ge abc$$
Check the condition for equality of AM-GM inequality to hold to decide the value of $a, b$ and $c$.