10-5-1. Among $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^{\circ}$, and the remaining one is less than $150^{\circ}$. For which $n$ is this possible? List all possible answers.
Answer: $8,9,10,11$.

Solution variant 1. Let the remaining angle be $x^{\circ}$. Using the fact that the sum of the angles of an $n$-sided polygon is $180^{\circ}(n-2)$, we get

$$
\begin{gathered}
150(n-1)+x=180(n-2) \\
n=\frac{x+120}{30}
\end{gathered}
$$

Since $9<x<150$, then $7<n<12$, so $n$ can only be $8,9,10,11$.

To complete the solution, we need to provide examples showing that all four situations are indeed possible. These examples are shown in the figures below.
![](https://cdn.mathpix.com/cropped/2024_05_06_4212ab26029b1cb1f16eg-04.jpg?height=368&width=1798&top_left_y=2282&top_left_x=171)