67th NMO Selection Tests for BMO and IMO

Since the number of positive divisors of a positive integer $m$ does not exceed $2\sqrt{m}$, and $a_1,\dots ,a_k$ are pairwise distinct positive divisors of $[a_1,\dots ,a_k]$, it follows that $[a_1,\dots ,a_k]\ge k^2/4$. Consequently, $$\begin{array}{rl}\sum _{k=1}^n\frac{1}{[a_1,\dots ,a_k]}& =\frac{1}{a_1}+\sum _{k=2}^n\frac{1}{[a_1,\dots ,a_k]}\le 1+\sum _{k=2}^n\frac{4}{k^2}<1+4\sum _{k=2}^n\frac{1}{k^2-\frac{1}{4}}\\ & =1+4\cdot 2\left(\frac{1}{3}-\frac{1}{2n+1}\right)<\frac{11}{3}<4.\end{array}$$