Difference between revisions of "1997 OIM Problems/Problem 6"
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== Problem == | == Problem == | ||
− | Let <math>\textbf{P} = { | + | 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 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
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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