Problem 1. The answer is (B).

The number $1000^{1000}$ is equal to $\left(10^{3}\right)^{1000}=10^{3000}$, which is written as 1 followed by 3000 zeros. The total number of digits is therefore 3001.

(Problem proposed by P. Francini)
1. Camilla is very patient and is writing out, in full, the entire number $1000^{1000}$. How many digits does she have to write in total?
(A) 1000
(B) 3001
(C) 1000001
(D) 1001
(E) 1004