A Diagonal of Two Squares
Two squares, with side lengths A and B, are placed together such that
the right side of square A touches the left side of square B and
their bases are collinear. A diagonal is drawn
from the bottom left corner of square A to the top right corner of
square B.

Find the area below the diagonal in square A, in terms of A and B.

Find the smallest integers A < B, such that the area from part 1 is
an integer. Repeat for A > B. Is there a restriction on the relationship
between A and B to achieve this?

Find A and B (A not equal to B) such that the diagonal is an integer
regardless of which square is on the left.
[I don't know if this can be solved with A and B both integers.]
Source: Original. Similar to
a problem at the
U. of Miss' Geometry Gambit Contest.
Solution
Mail to Ken