MHB Find the Maximal Value Challenge

[anemone]

It's given that $p+m+n=12$ and that $p, m, n$ are non-negative integers. What is the maximal value of $pmn+pm+pn+mn$?

 

[MarkFL]

My solution:

Spoiler

Given the cyclic symmetry of the variables (and the constraint), we know the extremum occurs for:

$$p=m=n=4$$

And so the objective function at these values is:

$$f(4,4,4)=4^3+3\cdot4^2=4^2(4+3)=112$$

Observing that:

$$f(3,4,5)=107$$

We take:

$$f_{\max}=112$$