9.9. On a plane, $N$ points are marked. Any three of them form a triangle, the angles of which in degrees are expressed by natural numbers. For what largest $N$ is this possible

$$
\text { (E. Bakayev) }
$$
Answer: 180.

![](https://cdn.mathpix.com/cropped/2024_05_06_f072938238320b6732abg-08.jpg?height=289&width=337&top_left_y=162&top_left_x=954)

Fig. 1

First Solution. Example. First, we will show that the required is possible when $N=180$. Mark 180 points on the circle, dividing it into 180 equal arcs, each $2^{\circ}$. The measure of any arc with endpoints at two of the marked points is expressed as an even number of degrees, so the measure of any inscribed angle in the circle, formed by three marked points, is a natural number of degrees. Therefore, 180 marked points satisfy the condition of the problem.

Estimate. It remains to prove that $N \leqslant 180$. Any three marked points form a triangle, so they cannot lie on the same line. Assuming the marked points are located on the coordinate plane, let $A$ be any of them with the maximum ordinate. Among the remaining points, choose points $B$ and $C$ such that the angle $B A C$ is maximized.

From the problem's condition, it follows that in triangle $A B C$, the measures of angles $A B C$ and $A C B$ are at least $1^{\circ}$, so the measure of angle $B A C$ is at most $178^{\circ}$. Due to the choice of points $B$ and $C$, the remaining $N-3$ marked points lie strictly inside angle $B A C$, and each ray starting from point $A$ contains no more than one of them. By drawing a ray through each marked point inside angle $B A C$ starting from point $A$, we obtain $N-3$ distinct rays, dividing $\angle B A C$ into $N-2$ angles. If $N-2>178$, then at least one of these angles has a measure less than $1^{\circ}$ and is an angle of some triangle with vertices at three marked points, which contradicts the problem's condition. Therefore, $N-2 \leqslant 178$, that is, $N \leqslant 180$, which was to be proved.

Remark 1. The choice of the point $A$ used in the solution can also be described in the following ways.