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numbers in a circle

Source: 2022 Centroamerican and Caribbean Mathematical Olympiad, P2

December 1, 2022
combinatoricsalgebra

Problem Statement

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers a,b,c,d,e,fa,b,c,d,e,f respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute 2022,20222022, 2022 replacements have been made and each person have in his paper it´s initial number. Find all the posible values of abc+defabc+def.
<spanclass=′latex−bold′>Note:</span><span class='latex-bold'>Note:</span> If at the beginning of the minute NN Ana, Beto, Carlos have the numbers x,y,zx,y,z, respectively, then at the end of the minute NN, Beto is going to have the number x+y+zx+y+z.
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