{
    "problem": "What is the greatest integer $n$ such that $n^2 - 11n +24 \\leq 0$?",
    "level": "Level 3",
    "type": "Algebra",
    "solution": "We can factor $n^2-11n+24$ as $(n-3)(n-8)$. For this quantity to be less than or equal to 0, one of the factors must be less than or equal to 0 and the other factor must be greater than or equal to 0. Specifically, since $n-8<n-3$ for all $n$, we must have $$n-8 \\le 0 \\le n-3.$$ The first inequality, $n-8\\le 0$, tells us that $n\\le 8$. The second inequality, $0\\le n-3$, tells us that $n\\ge 3$. The solutions to the original inequality must satisfy both conditions, so they are given by $3\\le n\\le 8$. The largest integer in this interval is $n=\\boxed{8}$."
}