Brazilian Math Olympiad

a. The erased numbers were $2=2\cdot 1$, $4=2\cdot 2$, $\dots$, $2010=2\cdot 1005$. So $2011-1005=1006$ numbers were left on the board.

b. We can list the numbers: they are $1$, $11$, $101$, $111$, $1001$, $1011$, $1101$, $1111$, a total of $8$.

OR we can argue that the number is of the form (abc1), where $a$, $b$, $c$ are digits equal to either $0$ or $1$. Notice that the units digit must be $1$.