In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $20$. What is the greatest possible perimeter of the triangle?

Let sides be $x,3x,20$. Triangle inequality: $x+20>3x\Rightarrow x<10$; $x+3x>20\Rightarrow x>5$.

So $x\in\{6,7,8,9\}$; greatest perimeter at $x=9$: sides $9,27,20$, perimeter $=\boxed{56}$.