2. On one website, there is a poll to determine the best hockey player of the last decade. The number of votes for each player is given as a percentage, rounded to the nearest whole number. After Mirek voted for Jaromír Jágr, his share of $7 \%$ did not change. What is the minimum number of people, including Mirek, who voted? We assume that each participant in the poll voted exactly once, and for only one player.
2. Let $p$ be the number of participants in the survey, including Miro, and $j$ be the number of votes for Jágar. The $7\%$ can be rounded from numbers in the interval $\langle 6.5\% ; 7.5\%)$ or $\langle 0.065 ; 0.075)$. Before Miro's vote, Jágar had $j-1$ votes and after, he had $j$ votes. Therefore, it must hold that

$$
0.065 \leqq \frac{j-1}{p-1} < 0.075
$$

Multiplying the entire inequality by $p-1$ gives:

$$
0.065(p-1) \leqq j-1 < 0.075(p-1)
$$

Adding 1 to all parts of the inequality:

$$
0.065(p-1) + 1 \leqq j < 0.075(p-1) + 1
$$

This can be rewritten as:

$$
0.065p + 0.935 \leqq j < 0.075p + 0.935 \quad \text{(1)}
$$

Since $j$ must be an integer, the difference between the upper and lower bounds must be at least 1:

$$
0.075p + 0.935 - (0.065p + 0.935) \geqq 1
$$

Simplifying this:

$$
0.01p \geqq 1
$$

Thus:

$$
p \geqq 100
$$

However, we need to find the smallest $p$ such that there exists an integer $j$ satisfying the inequalities (1). From the condition $p \geqq 94$, we get:

$$
j \geqq 0.065 \cdot 94 + 0.935 = 7.045
$$

Thus:

$$
j \geqq 8
$$

From the inequality $j < 0.075p + 0.935$, we get:

$$
p > \frac{j - 0.935}{0.075}
$$

For $j = 8$:

$$
p > \frac{8 - 0.935}{0.075} = \frac{7.065}{0.075} \approx 94.2
$$

Since $p$ must be an integer, we have:

$$
p \geqq 95
$$

However, we need to check if $p = 95$ satisfies the first inequality:

$$
0.065 \cdot 95 + 0.935 \leqq j < 0.075 \cdot 95 + 0.935
$$

$$
6.175 + 0.935 \leqq j < 7.125 + 0.935
$$

$$
7.11 \leqq j < 8.06
$$

Since $j$ must be an integer, $j = 8$. Now we check if $p = 95$ satisfies the second inequality:

$$
95 \leqq \frac{8 - 0.935}{0.065}
$$

$$
95 \leqq \frac{7.065}{0.065} \approx 108.69
$$

This is not true, so $p = 95$ does not work. We need to find the next integer $p$ that satisfies both inequalities. From the inequality:

$$
p > \frac{320}{3} \approx 106.67
$$

Thus:

$$
p \geqq 107
$$

We need to check if $p = 107$ works for $j = 8$:

$$
0.065 \cdot 107 + 0.935 \leqq j < 0.075 \cdot 107 + 0.935
$$

$$
6.955 + 0.935 \leqq j < 8.025 + 0.935
$$

$$
7.89 \leqq j < 8.96
$$

Since $j$ must be an integer, $j = 8$. Now we check if $p = 107$ satisfies the second inequality:

$$
107 \leqq \frac{8 - 0.935}{0.065}
$$

$$
107 \leqq \frac{7.065}{0.065} \approx 108.69
$$

This is true, so $p = 107$ is the smallest $p$ that satisfies both inequalities.

For a complete solution, award 6 points. For deriving a system of two inequalities analogous to system (1), award 2 points. For the step-by-step derivation of estimates for the integers $p$ and $j$, award 3 points. For verifying that $p = 107$ is the smallest $p$, award 1 point.