Task 3.

## Maximum 10 points

In the Country of Wonders, a pre-election campaign is being held for the position of the best tea lover, in which the Mad Hatter, March Hare, and Dormouse are participating. According to a survey, $20 \%$ of the residents plan to vote for the Mad Hatter, $25 \%$ for the March Hare, and $30 \%$ for the Dormouse. The rest of the residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter must attract to ensure he does not lose to the March Hare and the Dormouse (under any distribution of votes), knowing that each of the undecided voters will vote for one of the candidates. The winner is determined by a simple majority of votes. Justify your answer.
# Solution:

Let the number of residents in Wonderland be $N$, then $0.2 N$ residents are going to vote for Dum, $0.25 N$ residents for the Rabbit, and $0.3 N$ residents for Sonya. The undecided voters are $0.25 N$ residents. Let $\alpha$ be the fraction of the undecided voters who are going to vote for Dum. Dum will not lose the election if the number of votes for him is not less than the number of votes for his opponents. He will receive $0.2 N + 0.25 N \alpha$ votes. In the "worst" case, the remaining fraction $1 - \alpha$ of the undecided voters will vote for the Rabbit or Sonya. Therefore, for the condition of the problem to be met, the following must hold:

$$
\left\{\begin{array} { c } 
{ 0.2 N + 0.25 N \alpha \geq 0.25 N + 0.25 N (1 - \alpha), } \\
{ 0.2 N + 0.25 N \alpha \geq 0.3 N + 0.25 N (1 - \alpha). }
\end{array} \Leftrightarrow \left\{\begin{array}{c}
0.2 + 0.25 \alpha \geq 0.25 + 0.25(1 - \alpha) \\
0.2 + 0.25 \alpha \geq 0.3 + 0.25(1 - \alpha)
\end{array}\right.\right.
$$

It is clear that it is sufficient to solve the second inequality of the system:

$$
0.2 + 0.25 \alpha \geq 0.3 + 0.25 - 0.25 \alpha \Leftrightarrow 0.5 \alpha \geq 0.35 \Leftrightarrow \alpha \geq 0.7
$$

The smallest value of $\alpha = 0.7$, which corresponds to $70\%$ of the undecided voters.

Answer: $70\%$.

## Grading Criteria:

- 10 points - the correct answer is obtained with proper justification.
- 5 points - there is a correct sequence of reasoning leading to the right answer, but an arithmetic error has been made.
- 0 points - the solution does not meet any of the criteria provided.