Let x & y be positive real numbers such that x^3 + y^3 + 1/27 = xy. Find the value of 1/x.

x^3+y^3+1/27 = xy
=> x^3 + y^3 + (1/3)^3 = 3 x y (1/3)
if a^3 + b^3 + c^3 = 3abc then a + b + c = 0 or a=b= c

as (a^3+b^3+c^3) -3abc = 1/2(a+b+c) ((a-b)^2 + (b-c)^2 + (c-a)^2) 


so x + y + 1/3 = 0 or x = y= 1/3


as x+y+1/3 cannot be zero for x and y positive so x= y = 1/3 

or 1/x = 3