SAUDI ARABIAN MATHEMATICAL COMPETITIONS

Let $a_1,a_2,\dots ,a_{19}$ be the given numbers and suppose on the contrary that the set $S$ of all $\left(\genfrac{}{}{0ex}{}{19}{2}\right)=171$ numbers, which are written on the paper, are all different, $$d_1<d_2<\dots <d_{171}$$ Denote $k$ as the number of even values among the $19$ given numbers, and $t$ as the number of even values in $S$. It is easy to see that for any $d\in S$, there exist $a_x,a_y$ such that $d=\gcd (a_x,a_y)$; and this number $d$ is even if and only if $a_x,a_y$ are both even. Hence, we have $t=\left(\genfrac{}{}{0ex}{}{k}{2}\right)$.

Then the number of odd values in $S$ is $171-t$. Since $d_{171}-d_1<180$ then the number of even values and odd values in $S$ does not exceed $90$, which implies that $$\{\begin{array}{l}t\le 90\\ 171-t\le 90\end{array}\text{ so }81\le (\genfrac{}{}{0ex}{}{k}{2})\le 90.$$ Note that $$\left(\genfrac{}{}{0ex}{}{13}{2}\right)=78<91=\left(\genfrac{}{}{0ex}{}{14}{2}\right)$$ so there does not exist positive integer $k$ such that $81\le \left(\genfrac{}{}{0ex}{}{k}{2}\right)\le 90$, contradiction. Hence, the numbers written on the paper cannot be all different.