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Geometry · Distances on a line · Parity · Proof by contradiction

Problem 1, 2022

CityProof

Exactly \(2021\) points are chosen on the line \(AB\), and none of them lies on the segment \(AB\). Prove that the sum of the distances from these \(2021\) points to \(A\) can never be equal to the sum of their distances to \(B\).

A B X1 X2 X3 X4 X5
Five of the \(2021\) chosen points. The thick segment \(AB\) is forbidden territory: every chosen point lies on the line but outside it.

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Serbian Municipal Competition 2022, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source

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