Answer
$q_A = \frac{Q}{4}$
$q_B = 0$
$q_C = \frac{Q}{4}$
Work Step by Step
Since the spheres are identical, when two spheres are brought in contact, the charges will arrange themselves so that both spheres have the same net charge.
Initially:
$q_A = Q$
$q_B = 0$
$q_C = 0$
After sphere A touches sphere B:
$q_A = \frac{Q}{2}$
$q_B = \frac{Q}{2}$
$q_C = 0$
Since sphere C is grounded when it is touching sphere B, electrons will flow through the ground connection so that the net charge on sphere B and the net charge on sphere C are both zero.
After sphere B touches sphere C:
$q_A = \frac{Q}{2}$
$q_B = 0$
$q_C = 0$
After sphere C touches sphere A:
$q_A = \frac{Q}{4}$
$q_B = 0$
$q_C = \frac{Q}{4}$
