A "dot" (commonly also called a "lattice point") is a point with integer coordinates.

In the plane, what is the total number of dots inside or on the boundary of the triangle with vertices (0,0), (x,0), (x,y) where x and y are positive integers?

In the event that it is not utterly obvious from the form of your answer that a whole number is being specified, give an independent argument to show this.

What total do you get if you count the three vertex dots together as just half a dot and any other boundary dots as half a dot each?

Since

[ (x+1)*(y+1) + gcd(x,y) + 1 ] / 2

represents the total number of points within or on the edge of the triangle, and the second parts asks for those points at the vertices to count only as 1 altogether, and those points elsewhere on the edge to count as 1/2,we just have to subtract out:

2 (for the three vertices counting as one)
(y-1)/2 (halving along the vertical edge, other than the vertices)
(x-1)/2 (halving along the horizontal edge, other than the vertices)
(gcd(x,y)-1)/2 (halving along the diagonal, NOT counting the two vertices)

So it comes out to

[ (x+1)*(y+1) - x - y ] / 2