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Found problems: 1

2023 USAMTS Problems, 3
Tags: usmats
We say that three numbers are balanced if either all three numbers are the same, or they are all different. A grid consisting of hexagons is presented in Figure 1. Each hexagon is filled with the number 1, 2, or 3, so that for any three hexagons that are mutually adjacent and oriented with two hexagons on the bottom and one hexagon on the top (as in Figure 3), the three numbers in the hexagons are balanced. Prove that when the grid is filled completely, the three numbers in the three shaded hexagons are balanced. (An example of a partially filled-in grid is shown in Figure 2. There are other ways of filling in the grid.)