Given is a planar set of 25 points such that among any three there exists a pair at the distance less than 1. Prove that there exists a circle of radius 1 that contains at least 13 of the given points.
Solution
Contact
Front page
Contents
Up
Copyright © 19962018 Alexander Bogomolny
Given is a planar set of 25 points such that among any three there exists a pair at the distance less than 1. Prove that there exists a circle of radius 1 that contains at least 13 of the given points.
Pick arbitrarily one point A out of the given 25 and consider a circle B(A, 1) centered at A and radius 1. If all the remaining points lie inside B(A, 1), there is nothing to prove  B(A,1) is the circle we've been looking for.
Else, there is a point B that lies outside B(A, 1). Let B(B, 1) be a unit circle centered at B. By definition, the distance between A and B exceeds 1. For any point C, by the given condition, either the distance to A or the distance to B is less then 1. In other words, each of the remaining points belongs to either B(A, 1) or B(B, 1). By the Pigeonhole Principle, at least 13 lie in the same circle.
Related material Read more...
 





200 points have been cosen on a circle, all with integer number of degrees. Prove that the points there are at least one pair of antipodes, i.e., the points 180° apart.




If each point of the plane is colored red or blue then there are two points of the same color at distance 1 from each other.




The integers 1, 2, ..., 10 are written on a circle, in any order. Show that there are 3 adjacent numbers whose sum is 17 or greater.




Given a planar set of 25 points such that among any three of them there exists a pair at the distance less than 1. Prove that there exists a circle of radius 1 that contains at least 13 of the given points.




Prove that among any five points selected inside an equilateral triangle with side equal to 1, there always exists a pair at the distance not greater than .5.




Let A be any set of 19 distinct integers chosen from the arithmetic progression 1, 4, 7,..., 100. Prove that there must be two distinct integers in A whose sum is 104.




Prove that in any set of 51 points inside a unit square, there are always three points that can be covered by a circle of radius 1/7.




Five points are chosen at the nodes of a square lattice (grid). Why is it certain that at least one midpoint of a line joining a pair of chosen points, is also a lattice point?




Prove that there exist two powers of 3 whose difference is divisible by 1997.




If 9 people are seated in a row of 12 chairs, then some consecutive set of 3 chairs are filled with people.




Given any sequence of n integers, positive or negative, not necessarily all different, some consecutive subsequence has the property that the sum of the members of the subsequence is a multiple of n.




In every polyhedron there is at least one pair of faces with the same number of sides.







Contact
Front page
Contents
Up
Copyright © 19962018 Alexander Bogomolny