Pick's Theorem

Rarely Tested

To find the number of integral points inside the closed area, we can use Pick's Theorem.
Pick's Theorem: $$A\ =\ I+\frac{B}{2}-1$$ (Where, A = Area of the closed figure, I = the number of points inside the figure, B = the number of points on the boundaries)

Question 1

What are the total number of points with integral coordinates inside the triangle (excluding points on the boundary) bounded by the vertices (0,0); (100,0) and (0,50)

Question 2

Find the number of points with integral coordinates which lie in the region enclosed by |x|+|y| = 4 or on the boundary.

Question 3

Consider a triangle drawn on the X-Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X,Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is