When is this an Integer?Algebra Level 5
Let \(0< x\le 1000\) be a real constant. Find the number of distinct \(x\)'s such that, for positive real variables \(a,b,c\) the expression above is an integer.
Note: \(a,b,c\) are variables, \(x\) is a constant. The minimum value is taken where \(a,b,c\) vary over positive reals.