11. Magic Pen (recommended for 8th grade, 1 point). Katya correctly solves a problem with a probability of $4 / 5$, while the magic pen correctly solves a problem without Katya's help with a probability of $1 / 2$. In the test, there are 20 problems, and to get a B, one needs to solve at least 13 of them correctly. How many problems should Katya solve on her own, and how many should she entrust to the magic pen, so that the expected number of correct answers is at least 13?
Solution. Let $x$ be the number of examples Katya solves herself, and $20-x$ be the number of examples solved by the pen. Then the expected number of correctly solved problems is

$$
\frac{4}{5} x+\frac{1}{2}(20-x)=0.3 x+10
$$

From the inequality $0.3 x+10 \geq 13$ we get that $x \geq 10$. Therefore, Katya needs to try to solve at least 10 examples on her own.

Answer: at least 10 examples.