Proof of Murphy-Cohen Conjecture on One-dimensional Hard Ball Systems
  We prove the Murphy and Cohen's conjecture that the maximum number of collisions of n + 1 elastic particles moving freely on a line is n(n+1)/2 if no interior particle has mass less than the arithmetic mean of the masses of its immediate neighbors. In fact, we prove the stronger result that, for the same conclusion, the condition no interior particle has mass less than the geometric mean, rather than the arithmetic mean, of the masses of its immediate neighbors suffices. 
Lizhou Chen