MathDB

10

Part of 2021 Irish Math Olympiad

Problems(1)

Minimum Distance in a Quarter Plane With Fixed Centroid

Source: Irish Mathematical Olympiad 2021 P10

5/9/2021
Let P1,P2,…,P2021P_{1}, P_{2}, \ldots, P_{2021} be 2021 points in the quarter plane {(x,y):x≥0,y≥0}\{(x, y): x \geq 0, y \geq 0\}. The centroid of these 2021 points lies at the point (1,1)(1,1).
Show that there are two distinct points Pi,PjP_{i}, P_{j} such that the distance from PiP_{i} to PjP_{j} is no more than 2/20\sqrt{2} / 20.
combinatoricscombinatorial geometrygeometryCentroid