How should you saw a block of stone of 40 kg into four pieces such that you can measure all weights of an integer from 1 kg to 40 kg (so 1 kg, 2 kg, 3 kg, 4 kg, ..... 39 kg and 40 kg) on a balance?

Suppose that 1 out of 1000 people of a specific country are infected with the deadly X-virus. An inhabitant of that country goes the hospital and gives some blood for a survey. Her blood is checked on the X-virus by a test which gives a correct outcome in 99% of the cases. The test points out that the woman is infected. After the investigation, a doctor tells the woman that he is 99% sure that she is infected with the X-virus. Can the doctor be right by making this worrying statement?

To enter the house of the professor, you first have to walk up a stairs which consists out of twenty steps. The professor thinks his stairs are boring so he decides to paint them. He buys two colors, yellow and green. Every step will be either painted yellow or green and moreover the professor does not want to have two yellow steps directly after each other. In how many ways can the professor paint his stairs? And what is the name of the professor?