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11st ibmo - costa rica 1996/q6.

Source: Spanish Communities

April 23, 2006

geometrycircumcircletrigonometrygeometry unsolved

Problem Statement

There are

n

different points

A_1, \ldots , A_n

in the plain and each point

A_i

it is assigned a real number

\lambda_i

distinct from zero in such way that

(\overline{A_i A_j})^2 = \lambda_i + \lambda_j

for all the

i

,

j

with

i\neq{}j

} Show that:
(1)

n \leq 4

(2) 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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