# What is the largest possible product of those $3$ number?

Suppose $n$ is a positive integer and $3$ arbitrary numbers are choosen from the set $\{ 1,2,3, \cdots , 3n+1 \}$ with their sum equal to $3n+1$. What is the largest possible product of those $3$ number ?

Let the numbers be $a,b$ and $c$. From AM-GM, you have $$\dfrac{a+b+c}3 \geq \sqrt[3]{abc} \implies abc \leq \left(\dfrac{3n+1}3 \right)^3$$where the optimum is attained when$a=b=c=\dfrac{3n+1}3 = n + \dfrac13$. For the numbers in your set, the optimum is attained when $a=b=n$ and $c=n+1$. Hence, $$\max\{abc\} = n^2(n+1)$$
If $a \ne b \ne c$ then what is the largest possible product of those $3$ number ?
I think may be $(n-1)n(n+2)$.