SOFIA, Bulgaria — More than 5,000 people, mostly young, rallied outside the Palace of Justice here on July 31 to condemn violence against women. “Not one more woman!” proclaimed many of the signs. “We will not be silent,” was the theme of the action at Bulgaria’s capital. “Stop the genocide against women.” Similar protests took place nationwide. Bulgarian National Television reported the evening of July 31 that many thousands filled the squares of more than 30 cities and towns across the country. Demonstrations with the same demands spread to other cities on August 1. A harrowing attack against an 18-year-old woman — identified so far only by her initials, DM, to protect her from further abuse — and the authorities’ initial refusal to prosecute the perpetrator prompted the actions.