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2011 BAMO 2 numbers in 5 circles in a row

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August 27, 2019
combinatoricsnumber theorySum

Problem Statement

Five circles in a row are each labeled with a positive integer. As shown in the diagram, each circle is connected to its adjacent neighbor(s). The integers must be chosen such that the sum of the digits of the neighbor(s) of a given circle is equal to the number labeling that point. In the example, the second number 23=(1+8)+(5+9)23 = (1+8)+(5+9), but the other four numbers do not have the needed value. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi9lL2M2MzVkMmMyYTRlZjliNWEzYWNkOTM2OGVmY2NkOGZmOWVkN2VmLnBuZw==&rn=MjAxMSBCQU1PIDIucG5n What is the smallest possible sum of the five numbers? How many possible arrangements of the five numbers have this sum? Justify your answers.
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