RO-NMO 2003 Problem 112

Let be eight real numbers $1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4.$ Prove that $$\begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \end{vmatrix} >0.$$ Marian Andronache, Ion Savu