MathDB

6

Part of 2009 Czech-Polish-Slovak Match

Problems(1)

At least 4n√n quadrilaterals with concurrent diagonals

Source: Czech-Polish-Slovak Match, 2009

8/21/2011
Let n≥16n\ge 16 be an integer, and consider the set of n2n^2 points in the plane: G={(x,y)∣x,y∈{1,2,…,n}}. G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}. Let AA be a subset of GG with at least 4nn4n\sqrt{n} elements. Prove that there are at least n2n^2 convex quadrilaterals whose vertices are in AA and all of whose diagonals pass through a fixed point.
geometryparallelogrampigeonhole principlecombinatorics unsolvedcombinatorics