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Let $A_1, A_2, \ldots, A_n$ be distinct points in the plane. Suppose that each point $A_i$ can be assigned a real number $\lambda_i \neq 0$ in such a way that
$A_i A_j^2=\lambda_i+\lambda_j, \quad \text { for all } i, j \text { with } i \neq j .$

(a) Show that $n \leq 4$.

(b) Prove that if $n=4$, then $\frac{1}{\lambda_1}+\frac{1}{\lambda_2}+\frac{1}{\lambda_3}+\frac{1}{\lambda_4}=0$.
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