# Task 1. Maximum 20 points

In a certain city, all residents can be divided into three conditional, equal in number groups: poor, middle, and rich. The income of the poor group constitutes $x$ percent of the total income of all city residents, the income of the middle group constitutes $3 x$ percent, and that of the rich - $6 x$ percent.

The city decided to introduce a tax on the income of the wealthy part of society at a rate of $\left(\frac{x^{2}}{5}+x\right)$ percent of their income. The collected tax amount is distributed as follows: two-thirds of the collected amount goes to the poor, one-third to the middle group. Calculate the incomes of the groups after the introduction of the tax (in \% of the total income of all city residents).

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# Solution:

First, we need to find 1 share $x$ by adding up all the shares of citizens.

We get: $6x + 3x + 1x = 10x$, from which we find that $1x$ equals $10\%$ ( $10x = 100\%, x = 10\%$)

Now we need to find the share of income of the citizens:

The share of income of the poor is: $1 * 10\% = 10\%$

The share of income of the middle group is: $3 * 10\% = 30\%$

The share of income of the rich is: $6 * 10\% = 60\%$

Let's calculate the tax amount:

$\left(\frac{x^2}{5} + x\right) = 100 / 5 + 10 = 20 + 10 = 30\%$

After the introduction of the tax, the income of the rich will be: $0.6 * 0.7 = 0.42\%$ of the total income of all residents.

The difference between $60\%$ and $42\%$, which is $18\%$, will be distributed among the remaining groups.

Thus, the income of the poor will be: $10\% + (18\% * (2 / 3)) = 10\% + 12\% = 22\%$

The income of the middle group will be: $30\% + (18\% * (1 / 3)) = 30\% + 6\% = 36\%$

Verification: $42\% + 22\% + 36\% = 100\%$

Answer: The income of the poor group: $22\%$, the income of the middle group: $36\%$, the income of the rich: $42\%$

## Criteria:

Maximum score for the problem is 20

20 points: The problem is solved correctly, clearly, and the solution process is detailed

10 points: The answer is obtained, but the solution process is not detailed

5 points: There is a generally correct chain of reasoning, but the problem is not fully solved

0 points: No solution