Question 44
Let $n\ge2$ be an integer. Let $a_1,a_2,\ldots,a_n\in\{1,2,\ldots,2n\}$ satisfy $\operatorname{lcm}(a_i,a_j)\gt2n$ for every $1\le i\lt j\le n$. Prove that
$$a_1a_2\cdots a_n\mid(n+1)(n+2)\cdots(2n-1)(2n).$$
Editorial correction: the source prints $1\le i\lt j\le2n$, although only $a_1,\ldots,a_n$ are defined. The upper index bound is corrected to $n$; the range of values $1,\ldots,2n$ is unchanged.
$$a_1a_2\cdots a_n\mid(n+1)(n+2)\cdots(2n-1)(2n).$$
Editorial correction: the source prints $1\le i\lt j\le2n$, although only $a_1,\ldots,a_n$ are defined. The upper index bound is corrected to $n$; the range of values $1,\ldots,2n$ is unchanged.