5. In the city of Udoyev, mayoral elections proceed as follows. If in a given round of voting no candidate receives more than half of the votes, then a subsequent round is held with the participation of all candidates except the one who received the fewest votes. (No two candidates ever receive the same number of votes; if a candidate receives more than half of the votes, they become the mayor and the election ends.) Each voter votes for one candidate in each round. If this candidate advances to the next round, the voter votes for them again. If the candidate is eliminated, all their voters vote for the same remaining candidate.

In the upcoming election, 2002 candidates ran. Ostap Bender became the mayor, finishing in $k$-th place in the first round by the number of votes. Determine the maximum possible value of $k$ if Ostap was elected a) in the 1002nd round; b)* in the 1001st round.
5. a) Ostap could not take the last, 2002nd place in the first round, as otherwise he would have been immediately eliminated from the list of candidates. Therefore, $k \leqslant 2001$.

Suppose all candidates in the first round received almost the same number of votes, Ostap took the second-to-last place, and in each subsequent round, he received all the votes of the eliminated candidate. Then Ostap will win at the moment when the number of eliminated candidates reaches half. This will happen precisely in the 1002nd round.

Let's perform an exact calculation in the case where the candidates in the first round received $10^{6}, 10^{6}+1, \ldots, 10^{6}+2001$ votes. Then in the 1001st round, Ostap still has less than half of the votes, specifically: the votes of all candidates who took the last 1001 places in the first round. However, in the 1002nd round, he already has more than half of all the votes. Indeed, in the 1002nd round, Ostap has

\[
\begin{array}{r}
10^{6}+\left(10^{6}+1\right)+\ldots+\left(10^{6}+1001\right)=1002 \cdot 10^{6}+\frac{1001 \cdot 1002}{2}= \\
=1002 \cdot 10^{6}+1001 \cdot 501=1002501501
\end{array}
\]

votes, and the total number of voters is

\[
\begin{aligned}
10^{6}+\left(10^{6}+1\right)+\ldots+ & \left(10^{6}+2001\right)=2002 \cdot 10^{6}+\frac{2001 \cdot 2002}{2}= \\
& =2002 \cdot 10^{6}+2001 \cdot 1001=2004003001
\end{aligned}
\]

It is easy to verify that this is less than twice the number of votes Ostap has.

b) Suppose $k>1$. Those who were eliminated in the first thousand rounds will be called underdogs, and all other candidates except Ostap will be called leaders. The candidate who took first place in the first round will be called the favorite. Since the number of underdogs is 1000 and the number of leaders is 1001, one of the leaders did not receive votes from the underdogs. In the first round (and later), he had more votes than any underdog (since in the end, the underdog was eliminated, not this leader). Therefore, the favorite also had more votes in the first round than any underdog. Hence, the favorite is a leader.

The maximum number of votes Ostap could have collected by the 1001st round is all the votes of the underdogs at the moment of their elimination and the votes of Ostap's initial voters. Any leader in any of the first thousand rounds (and especially in the 1001st) has more votes than the underdog of that round. The favorite definitely has more votes than Ostap had in the first round. Therefore, the leaders and the favorite together have more votes in the 1001st round than Ostap, and he cannot become the winner.

Comments. $1^{\circ}$. Additional question: Would the answer change in the problem if the votes of the eliminated candidate were arbitrarily divided among the remaining candidates?

$2^{\circ}$. Note that in part "a" we get a very large city (with more than two billion voters). It is interesting to understand how the answer depends on the number of voters. Note that from the condition, it follows that the city has at least

\[
0+1+\ldots+2001=2003001
\]

voters - otherwise, the number of votes for all candidates could not have been different.