A teacher proposes 80 problems to a student, informing that they will award five points for each problem solved correctly and deduct three points for each problem not solved or solved incorrectly. In the end, the student ends up with eight points. How many problems did he solve correctly?
Let $c$ be the number of problems solved correctly and $e$ be the sum of the number of problems solved incorrectly and problems not solved. Therefore, $c+e=80$ and $5 c-3 e$ is the number of points the student scored in the evaluation. In the present case,

$$
\left\{\begin{aligned}
c+e & =80 \\
5 c-3 e & =8
\end{aligned}\right.
$$

Solving the system, we find $c=31$ and $e=49$. Therefore, the student solved 31 problems correctly.