Difference between revisions of "1997 OIM Problems/Problem 6"
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Let <math>\textbf{P} = {P1, P2, \cdots , P_{1997}}</math> be a set of 1997 points inside a circle of radius 1, with <math>P_1</math> being the center of the circle. For each <math>k = 1, \cdots , 1997</math> let <math>x_k</math> be the distance from <math>P_k</math> to the point of <math>\textbf{P}</math> closest to <math>P_k</math> and different from <math>P_k</math>. Show that | Let <math>\textbf{P} = {P1, P2, \cdots , P_{1997}}</math> be a set of 1997 points inside a circle of radius 1, with <math>P_1</math> being the center of the circle. For each <math>k = 1, \cdots , 1997</math> let <math>x_k</math> be the distance from <math>P_k</math> to the point of <math>\textbf{P}</math> closest to <math>P_k</math> and different from <math>P_k</math>. Show that | ||
| − | <cmath>(x_1)^2 + (x_2)^2 + \cdots + (x_{1997})^2 | + | <cmath>(x_1)^2 + (x_2)^2 + \cdots + (x_{1997})^2 \le 9</cmath> |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Revision as of 14:37, 13 December 2023
Problem
Let 
be a set of 1997 points inside a circle of radius 1, with 
being the center of the circle. For each 
let 
be the distance from 
to the point of 
closest to and different from . Show that
![\[(x_1)^2 + (x_2)^2 + \cdots + (x_{1997})^2 \le 9\]](http://latex.artofproblemsolving.com/d/0/7/d073e37edb14bff92b025b159a8d973da115e443.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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