9.5. On the exam, each of the three students was given the same test consisting of 40 questions. We will call a question unmanageable if all answered it incorrectly; difficult if only one student answered correctly, easy if exactly two students answered correctly, and trivial if all three answered correctly. It is known that the number of easy questions is twice the number of unmanageable questions. What is the maximum number of unmanageable questions that the test can contain if the total number of correct answers is 64? Justify your answer.
Solution: Let $x_{3}, x_{2}, x_{1}$, and $x_{0}$ be the quantities of trivial, easy, difficult, and insurmountable questions, respectively. Then the condition of the problem can be written as a system:

$$
\left\{\begin{aligned}
x_{3}+x_{2}+x_{1}+x_{0} & =40 \\
3 x_{3}+2 x_{2}+x_{1} & =64 \\
x_{2} & =2 x_{0}
\end{aligned}\right.
$$

We are interested in the solutions of this system in non-negative integers. Express all variables in terms of $x_{0}$. Subtracting the first equation of the system from the second, we get the equality $2 x_{3}+x_{2}-x_{0}=24$. Taking into account the third equation of the system, we get $x_{3}=12-0.5 x_{0}$. Then from the first equation of the system we have:

$$
x_{1}=40-x_{3}-x_{2}-x_{0}=40-12+0.5 x_{0}-2 x_{0}-x_{0}=28-2.5 x_{0}
$$

For the last expression to be an integer, it is necessary and sufficient that the number $x_{0}$ be even, and for it to be non-negative, the inequality $x_{0} \leqslant \frac{28}{2.5}=11.2$ must be satisfied. The largest number that meets both conditions is 10. By checking, we find that when $x_{0}=10$, we get an admissible solution of the system: $x_{0}=10, x_{1}=3, x_{2}=20, x_{3}=7$.

Answer: The maximum number of insurmountable questions can be 10.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer with an example of the number of tasks of all difficulties | 7 points |
| Correct and justified answer without an example of the number of tasks of all difficulties | 6 points |
| The system of equations correctly describing the conditions of the problem is written out | 1 point |
| Correct answer without justification with an example of the number of tasks of all difficulties | 1 point |
| Correct answer without justification | 0 points |