SÃO PAULO, Brazil — Shaken by the biggest challenge to their authority in years, Brazil’s leaders made conciliatory gestures on Tuesday to try to defuse the protests engulfing the nation’s cities. But the demonstrators remained defiant, pouring into the streets by the thousands and venting their anger over political corruption, the high cost of living and huge public spending for the World Cup and the Olympics.

In a convulsion that has caught many in Brazil and beyond by surprise, waves of protesters denounced their leaders for dedicating so many resources to cultivating Brazil’s global image by building stadiums for international events, when basic services like education and health care remain woefully inadequate.

“I love soccer, but we need schools,” said Evaldir Cardoso, 48, a fireman at a protest here with his 7-month-old son.

The demonstrations initially began with a fury over a hike in bus fares, but as with many other protest movements in recent years — in Tunisia, Egypt or, most recently, Turkey — they quickly evolved into a much broader condemnation of the government.