Question 4
Find all real numbers $r$ such that, for every positive $a,b,c$,
$$\left(r+\frac a{b+c}\right)\left(r+\frac b{c+a}\right)\left(r+\frac c{a+b}\right)\ge\left(r+\frac12\right)^3.$$
$$\left(r+\frac a{b+c}\right)\left(r+\frac b{c+a}\right)\left(r+\frac c{a+b}\right)\ge\left(r+\frac12\right)^3.$$