Yatsenko I.V.

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. After the elections, it turned out that each voter voted for exactly one party (there were no invalid ballots, votes "against all" and the like) and each party received an integer number of seats. At the same time, the Party of Mathematics Enthusiasts received 25% of the votes. What is the maximum number of seats in the parliament that it could have received?
If 10 parties receive exactly 5% of the votes each, and two, including the Party of Mathematics Enthusiasts (PME), receive 25% each, then the representatives of PME will receive exactly 50 seats in the parliament.

Let's prove that PME cannot receive more seats. If 11 parties did not make it into the parliament, they would collectively receive no more than 55% of the votes, but $55 + 25 < 100$. Therefore, a maximum of 10 parties did not make it, and they collectively received no more than 50% of the votes. Thus, the parties that did make it collectively received at least 50% of the votes, and the 25% of PME will give it no more than half of the seats.

## Answer

50 seats.