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on Beat the average game (AlphaEvolve problem 39)

The statement

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.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.