[ Counting in two ways ]  [ Different tasks on cutting ]

Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies outside this triangle or is its vertex (such partitions are called triangulations). Find the number of triangles in the partition.
Calculate the sum of the angles of all triangles.

## Solution

The sum of the angles of triangles with a vertex at some vertex of the square is $90^{\circ}$, each of the 100 marked points contributes $360^{\circ}$. Since there are no other vertices of the triangles, the sum of the angles of all triangles in the partition is $100 \cdot 360^{\circ} + 4 \cdot 90^{\circ} = 202 \cdot 180^{\circ}$. Since the sum of the angles of a triangle is $180^{\circ}$, the number of triangles is 202.

## Problem