MathDB
12nd ibmo - mexico 1997/q6.

Source: Spanish Communities

April 22, 2006
geometrygeometry unsolved

Problem Statement

Let P={P1,P2,...,P1997}P = \{P_1, P_2, ..., P_{1997}\} be a set of 19971997 points in the interior of a circle of radius 1, where P1P_1 is the center of the circle. For each k=1.…,1997k=1.\ldots,1997, let xkx_k be the distance of PkP_k to the point of PP closer to PkP_k, but different from it. Show that (x1)2+(x2)2+...+(x1997)2≤9.(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.
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