If you have a beam perfectly balanced on a fulcrum and put a weight at the very end of one side, is there a way to know where you should put a heavier object on the other end?

yes there is a way to determine where to put the heavier weight on the other end.

Explanation:

For the beam to remain balanced the net torque on the beam must be zero, and this means the net torque on both sides must be the same.

torque = weight (force) x distance of the weight from the fulcrum

therefore

torque on the right side = torque on the left side

weight (force) on the left side x distance of the weight from the fulcrum =

weight (force) on the right side x distance of the weight from the fulcrum

therefore once we know the value of the two weights we want to put on the beam, and the distance of the less heavy weight from the fulcrum, we can substitute those values into the equation above to get the position we are to place the heavier weight to maintain balance.