Problem

Source: INMO 2021 Problem 3

Tags: geometry, construction, INMO



Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge? Note. A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points. Proposed by Prithwijit De and Sutanay Bhattacharya