AlphaEvolve problems·alphaevolve:39
Beat the average game
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Let $C$ denote the quantity $$C \\coloneqq \\sup_{\\mu} \\mathbb{P}[X_1 + X_2 + X_3 < 2X_4]$$ where $\\mu$ ranges over probability measures on $[0, \\infty)$ and let $X_1, \\ldots, X_4 \\sim \\mu$ are independent random variables with law $\\mu$. Establish upper and lower bounds on $C$ that are as strong as possible.
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qed.bot, “Beat the average game”, https://qed.bot/t/alphaevolve-39, as of 30 Sep 2026.