Large Head-on Collider

The contestants at Decoherence, the physics event of Pravega, are conducting an experiment. They place $M$ balls of mass $1$ kg and $N$ balls of mass $2$ kg on the x-axis so that the distances between two consecutive balls on the line are all equal, all the $1$ kg mass balls are on the right side of the origin and the $2$ kg mass balls are on the left side of the origin. Assume the balls all have