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Kosovo MO 2009 Grade 10, Problem 5

Source: Kosovo MO 2009 Grade 10, Problem 5

June 7, 2021
combinatorics

Problem Statement

In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be x1,x2,x3,x4x_1,x_2,x_3,x_4 such that x1+x2+x3+x4>0x_1+x_2+x_3+x_4>0. Now we define a game with these numbers: If one of them, i.e. xix_i, is a negative number, the player makes a move by adding the number xix_i to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.
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