Problem:
Suppose that y=43​x and xy=yx. The quantity x+y can be expressed as a rational number sr​, where r and s are relatively prime positive integers. Find r+s.
Solution:
The conditions imply that
(43​x)x=x43​x
and hence ±43​x=x43​, or ±43​=x−41​. Thus x=(±34​)4=81256​, and y=2764​. Then
x+y=81256​+2764​=81256+192​=81448​,
and the requested sum is 529​ .
The positive rational solutions to xy=yx are precisely
{(xn​,yn​)}={((1+n1​)n,(1+n1​)n+1)}
for positive integers n.
The problems on this page are the property of the MAA's American Mathematics Competitions