Problem:
Suppose that y=43x 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
(43x)x=x43x
and hence ±43x=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