1996 OIM Problems/Problem 6

Problem

There are $n$ different points $A_1, \cdots , A_n$ in the plane and each point $A_i$ has been assigned a real number $\lambda _i$ other than zero, so that

$(\overline{A_iA_j})^2 = \lambda _i + \lambda _j$

$\text{for all } i, j \text{ with } i \ne j\text{.}$

Show that

a) $n \le 4$

b) If $n=4$ , then $\frac{1}{\lambda _1}+\frac{1}{\lambda _2}+\frac{1}{\lambda _3}+\frac{1}{\lambda _4}=0$

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1996 OIM Problems/Problem 6

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