US-TST 2010 Problem 9

Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and $\binom{3k}{k} \equiv 1 \pmod{p}.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.