3. In a certain state, only liars and economists live (liars always lie, while economists tell the truth). At a certain moment, the state decided to carry out monetary and credit, as well as budgetary and tax reforms. Since it was unknown what the residents expected, everyone was asked several questions (with only "yes" and "no" answers). The questions were answered affirmatively as follows:

"Will taxes be raised?" $40 \%$,

"Will there be an increase in the money supply?" $30 \%$,

"Will bonds be issued?" $50 \%$,

"Will the gold and foreign exchange reserves be squandered?" $\mathbf{0} \%$.

What percentage of the population actually expects taxes to be raised, if each of them realistically expects only one measure? (14 points).
# Solution:

Let $\boldsymbol{x}$ be the proportion of liars in the country, then (1-x) is the proportion of economists. Each economist answers affirmatively to one question, and each liar answers affirmatively to three. Therefore, we can set up the equation:

$3 x+1-x=0.4+0.3+0.5+0 ; 2 x=0.2 ; x=0.1$. Thus, in the country, $10\%$ are liars and $90\%$ are economists. Since no one said they expect a depletion of gold and foreign exchange reserves, this means that all liars are expecting it. Therefore, the actual percentage of the population expecting a tax increase is $40-10=30\%$.

Answer: $30\%$.

| Points | Criteria for evaluating the performance of task № $\mathbf{3}$ |
| :---: | :--- |
| $\mathbf{14}$ | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are correctly performed. The correct answer is obtained. |
| $\mathbf{10}$ | A correct sequence of all steps of the solution is provided. There are gaps in justifying the percentage of the population that actually expects a tax increase, or a computational error or typo that does not affect the further course of the solution. As a result of this error or typo, an incorrect answer may be obtained. |
| $\mathbf{5}$ | The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is correctly performed, possibly inaccurately (for example, an equation is set up); - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the percentage of the population that actually expects a tax increase is found incorrectly). The solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the criteria for scoring 5 and 10 points. |