If `a, b, c` are positive real numbers such that `a + b + c = 1`, then prove that `a/(b + c)+b/(c+a) + c/(a+b) >= 3/2`

Explanation

Arithmetic mean of `(-1)`th powers `le (-1)`th power of arithmetic mean
`((((b+c)/(a+b+c))^(-1)+((c+a)/(a+b+c))^(-1)+((a+b)/(a+b+c))^(-1))/(3))ge (((b+c)/(a+b+c)+(c+a)/(a+b+c)+(a+b)/(a+b+c))/(3))^(-1)`
`implies ((a+b+c)/(b+c)+(a+b+c)/(c+a)+(a+b+c)/(a+b))/(3)ge ((2)/(3))^(-1)`
`implies (a)/(b+c)+1+(b)/(c+a)+1+(c)/(a+b)+1 ge (9)/(2)`
`implies (a)/(b+c)+(b)/(c+a)+(c)/(a+b) ge (9)/(2)-3`
or ` (a)/(b+c)+(b)/(c+a)+(c)/(a+b) ge (3)/(2)`.