Question 8
In a country there are $100$ electoral districts, all with the same number of voters. Each district elects one deputy to the national parliament among $3$ candidates representing the $3$ political parties $A$, $B$ and $C$. At the national level, parties $A$, $B$ and $C$ have the support of exactly $60\%$, $30\%$ and $10\%$ of the voters, but the distribution of supporters by district can be arbitrary.
The elections have two rounds. If in a district in the first round one of the three candidates obtains more than half of the votes, he wins the election and there is no second round. Otherwise, the two candidates with the most votes compete in the second round, where again the one who obtains more than half of the votes wins.
Each voter votes for the candidate of their favourite party. If that candidate is not in the second round, the voters of $A$, $B$ and $C$ vote for the candidates of $C$, $C$ and $B$, respectively.
What is the minimum number of districts that party $A$ wins? What is the maximum number of districts that $C$ could win?
**CLARIFICATION:** Assume that in no district does a candidate obtain exactly $50\%$ of the votes, and also that there are no candidates with the same number of votes.
The elections have two rounds. If in a district in the first round one of the three candidates obtains more than half of the votes, he wins the election and there is no second round. Otherwise, the two candidates with the most votes compete in the second round, where again the one who obtains more than half of the votes wins.
Each voter votes for the candidate of their favourite party. If that candidate is not in the second round, the voters of $A$, $B$ and $C$ vote for the candidates of $C$, $C$ and $B$, respectively.
What is the minimum number of districts that party $A$ wins? What is the maximum number of districts that $C$ could win?
**CLARIFICATION:** Assume that in no district does a candidate obtain exactly $50\%$ of the votes, and also that there are no candidates with the same number of votes.