Concursul Argument, 2012, Problema 1

[Horea Muresan]

Fie $a_{1},a_{2},\ldots,a_{n}$ si $b_{1},b_{2},\ldots,b_{n}$ numere reale distincte si consideram maricele coloana
$A_{1}=\left( \begin{matrix} 1\\ a_{1}\\ a_{1}^{2}\\ \vdots\\ a_{1}^{n} \end{matrix} \right) ,\ldots, A_{n}=\left( \begin{matrix} 1\\ a_{n}\\ a_{n}^{2}\\ \vdots\\ a_{n}^{n} \end{matrix} \right)$;
$B_{1}=\left( \begin{matrix} 1\\ b_{1}\\ b_{1}^{2}\\ \vdots\\ b_{1}^{n} \end{matrix} \right) ,\ldots, B_{n}=\left( \begin{matrix} 1\\ b_{n}\\ b_{n}^{2}\\ \vdots\\ b_{n}^{n} \end{matrix} \right)$

Sa se arate ca daca $\det(A_{1},A_{2},\ldots,A_{n},B_{i}-B_{j})=0, \forall i,j=\overline{1,n}$ atunci

$\det(B_{1},B_{2},\ldots,B_{n},A_{i}-A_{j})=0, \forall i,j=\overline{1,n}$

(am notat cu $(A_{1},A_{2},\ldots,A_{n},B_{i}-B_{j})$ matricea patratica de ordin $n+1$ cu coloanele
$A_{1},A_{2},\ldots,A_{n}$ si $B_{i}-B_{j}$).