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Lattice points -- Paths

Source: Central American Olympiad 2002, problem 6

December 30, 2009
geometrycalculusintegrationsymmetry

Problem Statement

A path from (0,0) (0,0) to (n,n) (n,n) on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its 2n\plus{}1 vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices (0,0) (0,0), (n,0) (n,0), (n,n) (n,n), (0,n) (0,n) into two parts with equal area.
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