SOLUTION: Which is the greatest/largest number x so that {{{ (a^2+a+1)(b^2+b+1)(c^2+c+1) >= xabc }}}

I think this problem should/must be reformulated in this way:

  ----------------------------------------------------------------
  Find the greatest/largest number x so that 
      (1)
  for all real positive a, b and c.
  ----------------------------------------------------------------

Solution

By completing the square you get

 = .

So   is always >= 3a for positive a, and the equality is achieved at a=1.

Similarly,   is always >= 3b  for positive b, and the equality is achieved at b=1;

             is always >= 3c for positive c, and the equality is achieved at c=1.


Therefore, x=13*3*3 = 27 in (1) provides that (1) is always true, and you can not use greater x.

Thus the answer is: x=27.