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2025 SSMO Relay Round 2 Problems/Problem 2

Problem

Let $T = TNYWR.$ Let $x_1,x_2,\dots, x_{T}$ be an increasing sequence of positive integers such that for every positive integer $1\le n \le T,$ the sum $x_1+x_2+\cdots + x_n$ is a multiple of $n$. Find the smallest possible value of $x_{T} - x_1$.

Solution

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