Let a,b,c be positive real numbers such that a+b+c=1. Find the minimum value of a+2b1+b+2c1+c+2a1.
Solution — click to reveal
By AM-HM, 3(a+2b)+(b+2c)+(c+2a)≥a+2b1+b+2c1+c+2a13,so a+2b1+b+2c1+c+2a1≥3a+3b+3c9=39=3.Equality occurs when a=b=c=31, so the minimum value is 3.