2. Idea of the solution: The Party of Mathematics Enthusiasts (PME) will receive the maximum number of seats in parliament if the total number of votes cast for non-qualifying (i.e., receiving no more than $5 \%$ of the votes) parties is maximized.

If 10 parties receive exactly $5 \%$ of the votes each, and two, including the Party of Mathematics Enthusiasts (PME), receive $25 \%$ each, then only two parties will enter parliament. Each of them will receive exactly 50 seats in parliament.

We will prove that PME cannot receive more seats. If 11 parties did not qualify for parliament, then together they would have received no more than $55 \%$ of the votes, but $55 \% + 25 \% < 100 \%$. Therefore, a maximum of 10 parties did not qualify for parliament, and they collectively received no more than $50 \%$ of the votes.

Thus, the parties that qualified for parliament received at least $50 \%$ of the votes. Since PME received $25 \%$ of the votes, it received no more than half of the seats in parliament, i.e., no more than 50 seats.
2. In the elections to the 100-seat parliament, 12 parties participated. Parties that received strictly more than $5 \%$ of the voters' votes enter the parliament. Among the parties that entered the parliament, seats are distributed proportionally to the number of votes they received (i.e., if one party received $x$ times more votes than another, it will also receive $x$ times more seats in the parliament). After the elections, it turned out that each voter voted for exactly one party (there were no invalid ballots, votes "against all," etc.) and each party received an integer number of seats. Moreover, the Party of Mathematics Enthusiasts received $25 \%$ of the votes. What is the maximum number of seats in the parliament that the Party of Mathematics Enthusiasts could have received? (Explain your answer.)