 Solution

The risk-free weight is $1-1^{t}w$, so a zero risk-free weight requires $1^{t}w=1$. Define $H=1^t{\Sigma }^{-1}d$. If $H\ne 0$, this condition gives $\lambda =1/H$ andThese are the requested tangent-portfolio mean and variance under the usual condition $H>0$. Then $\lambda >0$, the normalized portfolio lies on the upper efficient ray, and its Sharpe ratio is $\sqrt{A}$.

There is a real qualification if the data are allowed to be arbitrary. If $H=0$, the efficient risky direction has zero total risky weight, and no finite multiple makes its risky weights sum to one. If $H<0$, the formal normalized portfolio above has negative excess mean and lies on the lower, dominated branch; it is not an efficient tangent portfolio as described in the question. The sign of the normalized tangency portfolio must therefore be checked, rather than inferred from positive definiteness of $\Sigma$ alone.