“Of course I knew it was illegal. But if you don’t pay, your child will go nowhere.”

But critics say China’s state-run education system — promoted as the hallmark of Communist meritocracy — is being overrun by bribery and cronyism. Such corruption has broadened the gulf between the haves and have-nots as Chinese families see their hopes for the future sold to the highest bidder.

“Corruption is pervasive in every part of Chinese society, and education is no exception,” Mr. Li said.

It begins even before the first day of school as the competition for admission to elite schools has created a lucrative side business for school officials and those connected to them.