36. Each side of an equilateral triangle is divided into 30 equal parts. Lines drawn through the division points parallel to the sides of the triangle divide it into 900 small triangles. What is the maximum number of vertices of the partition, no two of which lie on the same drawn line or side?
72.36. Answer. 21. Indeed, for each vertex, we can define three coordinates: the numbers of lines parallel to one of the sides of the triangle, counted from the vertex not belonging to the corresponding side (see Fig. 45). It is clear that the sum of the coordinates of any vertex is 30. Since the first (as well as the second or third) coordinates of any two marked vertices must be different, their sum is no less than \(0+1+2+\ldots+(n-1)=n(n-1) / 2\), and the sum of all coordinates is no less than \(3 n(n-1) / 2\), where \(n\) is the number of all marked vertices of the partition. But, on the other hand, this sum is equal to \(30 n\). Therefore, \(30 n \geqslant 3 n(n-1) / 2\), or \(n-1 \leqslant 20\). An example with 21 vertices is left to the reader as an exercise.