6 problems

Mean-value conjecture. As $p$ tends to infinity,

Residue distribution conjecture. As the prime modulus $p$ tends to infinity,

McNamara–Sagan's conjecture. The sequence $\{\binom{n+mu}{mv}\}_{m\geq 0}$ is infinitely log-concave for all $n\geq 0$ if and only if $u<v$ or $v=0$.

McNamara–Sagan's conjecture. The sequence

Stephan's third ratio conjecture.

Let $\binom{n}{k}$ be a fixed element of Pascal's triangle, and let a ray cross a sequence of entries in the triangle. Ray-unimodality conjecture. The sequence of binomial coeffici…