Bakayev E.B.

On a line, 100 points are marked, and one more point is marked outside the line. Consider all triangles with vertices at these points.

What is the maximum number of them that can be isosceles?
Let $l$ be a given line, $O$ be a marked point outside it, and $O H$ be the perpendicular dropped to $l$.

Estimate. The base of an isosceles triangle can lie on $l$ or not. The triangles of the first type are symmetric with respect to $O H$, so there are no more than 50 - half of the marked points on $l$. Triangles of the second type with a given base $O A$ can be no more than one, since the vertex is determined by the intersection of $l$ and the perpendicular bisector of $O A$. Therefore, there are no more than 100 triangles of the second type.

Example 1. From the ray $O H$, lay off rays at an angle of $18^{\circ}$. They will intersect $l$ at points $A_{1}$ and $B_{1}$, mark them. On $l$, in the other direction from $A_{1} B_{1}$, lay off the segment

$A_{1} A_{2}=A_{1} O$, then the segment $A_{2} A_{3}=A_{2} O, \ldots$, the segment $A_{49} A_{50}=A_{49} O$. Similarly, mark the points $B_{2}, \ldots, B_{50}$. There will be 50 isosceles triangles $O A_{i} B_{i}$, 49 triangles $O A_{i} A_{i+1}$ and $O B_{i} B_{i+1}$, and triangles $O A_{1} B_{2}$ and $O B_{1} A_{2}$ with angles $72^{\circ}, 72^{\circ}, 36^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f3ff818e52eab4ca192bg-45.jpg?height=231&width=706&top_left_y=639&top_left_x=676)

Example 2. Consider the vertices of a regular 101-gon. Mark one of them as $O$, its neighbors as $A$ and $B$, and for each of the other vertices $X$, mark its image $X^{\prime}$ under inversion with center $O$ and radius $O A$. Then all the marked points, except $O$, lie on $A B$ (with the rays to them dividing the angle $A O B$ into 99 equal angles). Since under inversion triangles $O X Y$ and $O Y^{\prime} X^{\prime}$ are similar, the number of isosceles triangles with vertices at the marked points is equal to the number of isosceles triangles of the form $O X Y$, where $X$ and $Y$ are vertices of the 101-gon. There are 50 such triangles with a base not containing $O$, and 100 with a base containing $O$.

## Answer

150 triangles.