Suppose that there are two assets, the first with initial price $$20$ and the second with intial price $$10.$ The two assets have just two possible values for their price at time $1,$ as indicated in the table below.

values $of(A_1,A_2)$:$,($$$22,$$$13\mathrm{.}50),($$$19,$$$4\mathrm{.}50)$

probability:$\frac{2}{5},\frac{3}{5}$

Assume that it is possible to sell short, that is$,$ to hold a negative proportion of wealth on one asset $($essentially selling it without owning it in order to invest extra money in the other asset$).$ What portfolio weights would result in a rate of return of $\mathrm{.}05$ on the portfolio regardless of which of the two outcomes happens?