## Task A-4.2.

Exactly three interior angles of a convex polygon are obtuse. What is the maximum number of sides that this polygon can have?
## First solution.

The sum of the exterior angles of a convex polygon is $360^{\circ}$.

Three exterior angles of the observed polygon are acute (opposite to the obtuse interior angles), while all other exterior angles must be obtuse or right.

2 points

At most three exterior angles can be obtuse or right, otherwise their sum would be $360^{\circ}$ or more, which is impossible.

This means that at most three interior angles are acute or right, so the observed polygon has at most six vertices.

We still need to check if there exists a hexagon with the described properties.

One such hexagon has angles of $90^{\circ}, 150^{\circ}, 90^{\circ}, 150^{\circ}, 90^{\circ}, 150^{\circ}$, and all sides are congruent.

![](https://cdn.mathpix.com/cropped/2024_05_30_44b5caa35f0d2fab7d89g-16.jpg?height=563&width=563&top_left_y=889&top_left_x=678)

This shows that the maximum number of sides of a polygon with the given properties is 6.