Answer
Since we do not know that value of $\frac{m_B}{m_A}$, we can not determine the speed of A.
Work Step by Step
We can find an expression for the x-component of the velocity of ball A after the collision:
$m_A~v_x = m_B~v$
$v_x = \frac{m_B}{m_A}~v$
We can find an expression for the y-component of the velocity of ball A after the collision:
$m_A~v_y = m_B~\frac{v}{2}$
$v_y = \frac{m_B}{m_A}~\frac{v}{2}$
We can find an expression for the speed of ball A:
$v_A = \sqrt{(\frac{m_B}{m_A}~v)^2+(\frac{m_B}{m_A}~\frac{v}{2})^2}$
$v_A = (1.12~\frac{m_B}{m_A})~v$
Since we do not know that value of $\frac{m_B}{m_A}$, we can not determine the speed of A.
