## Task 1 - 330841

Max is working through a number of problems in preparation for the Math Olympiad. When his friend Moritz asks him how many of these problems he has already solved and how many are still unsolved, he answers:

"The number of solved problems is 22 more than the number of unsolved problems. If you add the number of solved problems to three times the number of unsolved problems, you get a number that is less than 60. But if you add the number of solved problems to one third of the number of unsolved problems, you get a whole number that is greater than 30."

Determine whether these statements uniquely determine how many problems Max has worked on and how many he has solved! If this is the case, provide these numbers!
From the given information: If Max has solved $x$ problems, then he has not solved $x-22$ problems; it follows that

$$
\begin{aligned}
x+3 \cdot(x-22)30 \quad(2) \\
& x+7 \cdot(x-22) \text { is an integer }
\end{aligned}
$$

From (1) it follows that $x+3 x-6630$ and $x>28$.

Therefore, $x$ is one of the numbers 29, 30, 31, and consequently $x-22$ is one of the numbers 7, 8, 9.

Condition (3) requires that $x-22$ is divisible by 3; this is only satisfied by $x-22=9$ among the mentioned numbers.

Thus, it is uniquely determined from the given information: Max has solved $x=31$ problems out of a total of 40 problems.