Difference between revisions of "1997 OIM Problems/Problem 6"

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== Problem ==
 
== Problem ==
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
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Let <math>\textbf{P}  = {P_1, P_2, \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 \le 9</cmath>
 
<cmath>(x_1)^2 + (x_2)^2 + \cdots + (x_{1997})^2 \le 9</cmath>

Latest revision as of 06:54, 26 December 2023

Problem

Let $\textbf{P}  = {P_1, P_2, \cdots , P_{1997}}$ be a set of 1997 points inside a circle of radius 1, with $P_1$ being the center of the circle. For each $k = 1, \cdots , 1997$ let $x_k$ be the distance from $P_k$ to the point of $\textbf{P}$ closest to $P_k$ and different from $P_k$. Show that

\[(x_1)^2 + (x_2)^2 + \cdots + (x_{1997})^2 \le 9\]

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

Solution

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See also

https://www.oma.org.ar/enunciados/ibe12.htm