I am proving mathematically a mean variance type problem, which is as follows.Lets say we have n assets, with an expected retun matirx denoted as R (n X 1). Assuming that the sharpe ratios of all the assets are the same, ie for any asset we havee1 = SRvol1 + rf,where SR is the sharpe ratio and rf is the risk free. Let w be the matrix of weightsNow the tangent portfolio is given byMax (w'R - rf)/sqrt(w'Vw) - Lamba(1-w'I) wrt where w' is the transpose of the weight matrix and lambda is the lagrange multiplier take the constraint into account, it is required to prove that the solution does not change as a function of the sharpe ratio.

R = SRvol1 + rf, SRvol2 + rf SRvol3 + rf . . . SRvoln + rfand you want to maxT= (w'R - rf)/sqrt(w'Vw) - Lamba(1-w'I) wrt wSo, since R is the same for each, I think ( check this ) thatT simplifies to SR*(w'R*)/sqrt(w'V'w) - Lamba(1-w'I) where R* = vol1 + rf/SR vol2 + rf/SR . . . . . voln + rf/SRMaximizing c*AX where c is a constant, is the same as maximizing AX so SR on the outside doesn't effect the maximizationproblem,so we can now maximize T* = (w'R*)/sqrt(w'V'w) - Lamba(1-w'I) but, the same amount, c= rf/SR is being added to the vector R* each timeso, since maximizing w'(x + c) where c is a constant is the same as maximizing w'x so , wecan just maximize T** = (w'R**)/sqrt(w'V'w) - Lamba(1-w'I) where R** = vol1 vol2 . . . . . voln R** doesn't involve SR so we have shown that the maximization is independent of the value of SR.