Let $O$ a point in plane and $M$ be a set of points such that for every $A, B\in M$, $A\neq B$ the angle $AOB$ is proper, that means measure of $\angle AOB$ couldn't be $0^\circ$ or $180^\circ$ and has an integer measure.
Find the maximum number of elements of the set $M$.
I think that the answer is $180$. If I take $M=\{A_1,..., A_{180}\}$ with $\angle A_1OA_2,..., \angle A_{179} O A_{180}$ adjacents with measures $1^\circ$ then all the angles formed will be proper. But I don't know how to justify that it is the maximum.
First, let's draw lines passing through the origin, on which each point in $M$ lies.
Because the angles are integer, the angle between adjacent two lines is at least $1^\circ.$ In this case, we can at most draw $180$ such lines. According to the definition of $M$, we know that there can only be one point on each line. Thus, at most, we have $180$ points.
If we have one more point, according to pigeonhole principle, there must be one line with two points, which leads to contradiction.