Only if he started with $1000 - 1024n < 0, n \in \mathbb{Z}^+$ ingots. If Aladdin has $1000 + x$ ingots and asks the Genie for some, he will afterwards have $$\frac{2000 + x}{2} = 1000 + \frac{x}{2}$$ ingots. If this is to remain an integer 10 times, $x$ must have initially been divisible by $2^{10} = 1024$. Thus the starting amount must have been $1000 + 1024n, n \in \mathbb{Z}$. Then the final amount is $$1000 + n > 1000 + 1024n \Rightarrow n < 0$$ as mentioned above. However, this is always negative.