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?

The number of points on the diagonal
is the gcd(x,y)+1 (this is counting
the two vertices).

If we look at the number of points
in or on the corresponding rectangle 

  (0,0) (x,0) (x,y) (0,y)

it is (x+1)*(y+1).

If we subtract the number of points
on the diagonal, divide by two, and
add back the points on the diagonal
we get the number of points in or on
the triangle:

  [ (x+1)*(y+1) + gcd(x,y) + 1 ] / 2

   x     y    (x+1)*(y+1)  gcd(x,y)

  even  even      odd        even
  even  odd       even       odd
  odd   even      even       odd
  odd   odd       even       odd

The numerator in all cases is even.

Have not looked at the second part yet.