Problem:
Find x2+y2 if x and y are positive integers such that
xy+x+y=71 and x2y+xy2=880
Solution:
Let a=x+y and b=xy, and note that the given equations imply
a+b=71 and ab=880.
Solving simultaneously, one finds that {a,b}={16,55}; i.e., either
x+y=55 and xy=16(1)
or
x+y=16 and xy=55(2)
It is easy to check that (1) has no solution in integers while in (2) we have {x,y}={5,11}. Consequently, x2+y2=52+112=146​.
The problems on this page are the property of the MAA's American Mathematics Competitions