During the contest, I could prove that it will work because (maximum \ power \ of \ a \ warrior \ = \ 10^9 ) * (maximum \ number \ of \ warriors \ = \ 10^5 ) = 10^{14} < 10^{15}. So, even if I don’t consider the gain in health we can get after fighting with every subsequent warrior, power of 10^{15} is large enough to kill all the remaining warriors.
As dragonemperor noted above, any number greater than or equal to twice the maximum power among all the warriors will work.