Given pairs $(a_1,b_1)$, $(a_2,b_2),\ldots, (a_n,b_n)$ of non-negative real numbers such that for any real $x$ and $y$ the equality $$\sqrt{a_1x^2+b_1y^2}+\sqrt{a_2x^2+b_2y^2}+\ldots+\sqrt{a_nx^2+b_ny^2}=\sqrt{x^2+y^2}$$Prove that $a_1=b_1,a_2=b_2,\ldots$,$a_n=b_n$ A. Vaidzelevich