The value of acceleration due to gravity at some height is given by

g1 = g*((R/R+h)^2)

Where h is the height of the particle (Q, outside the earth).

Value of acceleration due to gravity inside the earth is given by

g1 = g*(1-d/R)

where d is the depth of the particle (P, inside the earth)

putting g1 = g/2 in both the cases, you will get value of h and d.

adding both h and d, you will get the maximum possible distance in terms of radius of earth.