Problem 2. The answer is $\mathbf{( E )}$.

The numbers between 1 and 500 have their double between 2 and 1000, and are therefore certainly written in the notebook after Giovanna's intervention. Numbers greater than 500 are written only if they are odd: in total, this amounts to 250 different numbers. Therefore, after Giovanna's intervention, Carlo's notebook will contain $500+250=750$ different numbers. 

(Problem proposed by P. Francini)
2. Carlo has written in his notebook the list of all positive integers from 1 to 1000 (inclusive). Giovanna erases the even numbers from the list and replaces each of them with their half. How many different numbers will be written in Carlo's notebook after Giovanna's intervention?
(A) 650
(B) 900
(C) 500
(D) 600
(E) 750