What is the minimum velocity with which a body of mass m must enter...

Let velocity at the lowest point is "u" and velocity at the top most point is "v"

At highest point $\dfrac{m{v}^{2}}{\text{R}} = mg ⇒ v = \sqrt{gR}$

now ${v}^{2} = {u}^{2} - 2g( 2R ) ⇒ {u}^{2} = 5gR$

Minimum velocity required is $u = \sqrt{5gR}$