Here are the completed sentences for each prompt:

1. **They believed that the Chinese invented paper.**
It is **believed that the Chinese invented paper.**
The Chinese **are believed to have invented paper.**

2. **People said that the President was killed by a mad man.**
It was **said that the President was killed by a mad man.**
The President **was said to have been killed by a mad man.**

3. **They know that David left New York last week.**
It is **known that David left New York last week.**
David **is known to have left New York last week.**

4. **People thought that John would leave for Hanoi.**
John **was thought to be leaving for Hanoi.**
It **was thought that John would leave for Hanoi.**