3. In the 100th year of Besmiki's tenure as the President of the Currency Authority, he decided to issue new gold coins. In this year, he put into circulation an unlimited number of gold coins with a face value of $2^{100}-1$ yuan. In the following year, he put into circulation an unlimited number of gold coins with a face value of $2^{101}-1$ yuan. This continued every year until the day when the face value of a newly issued gold coin equaled the sum of the face values of some gold coins issued in previous years, at which point he was dismissed. When did this situation occur in Besmiki's tenure as President?
3. It happens in the 200th year of Besmiki's presidency.

Assume that the described scenario occurs in the $k$-th year of Besmiki's presidency. Then,
$$
2^{k}-1=a_{1}+a_{2}+\cdots+a_{n}=N-n,
$$

where $N$ is the sum of some powers of 2, all of which are divisible by $2^{100}$.

Since $2^{k}$ is also divisible by $2^{100}$, it follows that $n-1$ is also divisible by $2^{100}$.
Clearly, $n>1$.
Thus, $n \geqslant 2^{100}+1$.
This implies
$$
2^{k}-1 \geqslant\left(2^{100}-1\right)\left(2^{100}+1\right) \geqslant 2^{200}-1 \text {. }
$$

This indicates that $k \geqslant 200$.
Therefore, Besmiki cannot be dismissed from the presidency before the 200th year.

However, from $\left(2^{100}-1\right)\left(2^{100}+1\right)=2^{200}-1$, we know that Besmiki is dismissed in the 200th year.