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For $\\delta>0$, we prove that \\[\n  \\mathbb{P}\\left(S<\\mathbb{E} S+\\delta\\right)\\ge b_{n,\\delta}, \\] where $b_{n,\\delta}=\\delta(n/(n+\\delta))^n$ for $0<\\delta<1$ and $b_{n,\\delta}=(1-1/(n+\\delta))^n$ for $\\delta\\ge1$. The bound is sharp for every $n$ and $\\delta\\ge 1$. In particular, since $b_{n,\\delta} \\ge e^{-1}$ for $\\delta \\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $\\delta\\ge 1$.\n  The proof is found by ChatGPT 5.6 Pro. 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