On the vertices of a polygon with sides integers are written, in such a way that the number written on each vertex has the same parity as the sum of the numbers written on the two adjacent vertices (that is, if the number on the vertex is even, the sum of the numbers appearing on the adjacent vertices is also even, while if the number is odd the sum is also odd). Which of the following statements is certainly true?
Progetto Olimpiadi della Matematica - Gara di Febbraio
Pick one
Solution
Solution:
The answer is . Indeed, observe that every vertex on which an odd number is written has exactly one neighbor marked with an odd number, hence the odd numbers occur in pairs (and their total number is therefore even, possibly zero).
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