2025 SSMO Relay Round 1 Problems/Problem 1

Problem

Let $x_1, x_2, \ldots, x_7$ be distinct integers such that the mean of $\{x_i,x_{i+1},x_{i+2}\}$ is an integer for all integers $1\le i\le 5$ . Find the minimum possible positive value of $x_7 - x_1$ .

Solution

From the given condition, we know that $3$ divides both $x_1+x_2+x_3$ and $x_2+x_3+x_4$ , so $3$ divides their difference, which is $x_4-x_1$ . Analogous reasoning shows that $3$ divides $x_7 - x_4$ as well. Hence, $3$ divides $(x_4 - x_1) + (x_7-x_4) = x_7 - x_1$ . This means that the smallest possible positive value of $x_7 - x_1$ is $3$ . This minimum is indeed achievable -- take $\{x_1, x_2, \dots, x_7\}$ to be $\{0,6,9,12,15,18,3\}$ , for example -- so the answer is $\boxed{3}$ .

~Sedro

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2025 SSMO Relay Round 1 Problems/Problem 1

Problem

Solution