Problem

Source: 2024 Israel TST Test 8 P1

Tags: factorial, number theory, floor function



For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\]Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.