Find the minimum value of real number $m$, such that inequality \[m(a^3+b^3+c^3) \ge 6(a^2+b^2+c^2)+1\] holds for all positive real numbers $a,b,c$ where $a+b+c=1$.
Problem
Source: China south east mathematical olympiad 2006 day2 problem 6
Tags: inequalities, inequalities unsolved