Whether there are infinitely many integers $a, b, n$ with $a, b \ge \varepsilon n$ such that $a!\cdot b!$ divides $n!\cdot(a+b-n)!$ while $a+b$ exceeds $n$ by more than $C\cdot\log n$.

Let $\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that $$a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!,$$ and $$C \log n < a + b - n < C' \log n ?$$

Note that the website currently displays a simpler (trivial) version of this problem because $a + b$ isn't assumed to be in the $n + O(\log n)$ regime.

Barreto and ChatGPT-5.2 have proved that, for any $0 < C_1 < C_2$, there are infinitely many $a, b, n$ with $b = n/2$, $a = n/2 + O(\log n)$, and $C_1 \log n < a + b - n < C_2 \log n$ such that $a! b! \mid n! (a + b - n)!$

This appears to answer the question in the spirit it was intended.

This was formalized in Lean by Alexeev using Aristotle.