First, let's construct the correlation matrix of $A$, $B$ and $C$. $$ \rho = \begin{pmatrix} 1 & 0.9 & 0.1\\ 0.9 & 1 & 0.8\\ 0.1 & 0.8 & 1\\ \end{pmatrix} $$ We can use the fact that all correlation matrices are positive semidefinite to solve this problem. A necessary condition for a matrix to be positive semidefinite is that the determinant should be greater than or equal to zero $$ \det(\rho) \geq 0. $$ So, we simply need to calculate the determinant of this matrix. The determinant evaluates to $$ \det(\rho) = -0.316 $$ and therefore it is not possible for $A$ and $C$ to have a correlation coefficient of $0.1$.