This is an exercise in Carlo Bourlet's textbook, LECONS D'ALGEBRE ELEMENTAIRE. Sove the equations,
$$ x_1(x_2+x_3+\cdots +x_n)+2\cdot 1 (x_1+x_2+\cdots +x_n)^2=9a^2 $$ $$ x_ 2(x_1+x_3+\cdots +x_n)+3\cdot 2 (x_1+x_2+\cdots +x_n)^2= 25a^2 $$ $$\cdots \cdots$$ $$x_ n(x_1+x_2+\cdots +x_{n-1})+(n+1)\cdot n(x_1+x_2+\cdots +x_n)^2= (2n+1)^2a^2 $$
I tried to wtite them as, $$ -x_1^2+x_1(x_1+x_2+x_3+\cdots +x_n)+2\cdot 1 (x_1+x_2+\cdots +x_n)^2=9a^2 $$ $$-x_2^2+x_2(x_1+x_2+x_3+\cdots +x_n)+3\cdot 2 (x_1+x_2+\cdots +x_n)^2= 25a^2 $$ $$\cdots \cdots$$ $$-x_n^2+x_n(x_1+x_2+x_3+\cdots +x_n)+(n+1)\cdot n(x_1+x_2+\cdots +x_n)^2= (2n+1)^2a^2 $$
Then if add them together, I get, $$ -(x^2_1+x^2_2+x^2_3+\cdots +x^2_n)+\Big[1+\sum_{i=1}^n(i^2+i)\Big] S^2=\sum_{i=1}^n(2i+1)^2a^2 $$
I do not know how to deal with the first term now. If it can be represented as function of $S$, then it will be solved.