MathDB

At least 4n√n quadrilaterals with concurrent diagonals

Source: Czech-Polish-Slovak Match, 2009

August 21, 2011

geometryparallelogrampigeonhole principlecombinatorics unsolvedcombinatorics

Problem Statement

Let

n\ge 16

be an integer, and consider the set of

n^2

points in the plane:

G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.

Let

A

be a subset of

G

with at least

4n\sqrt{n}

elements. Prove that there are at least

n^2

convex quadrilaterals whose vertices are in

A

and all of whose diagonals pass through a fixed point.