Let
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
,
![a_2](/media/m/4/0/1/401f4cdfec59fba73ae32fa6769c72cb.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
be distinct positive integers,
![n\ge 3](/media/m/f/2/6/f26f60f0cef07be122e851956e32cd1d.png)
. Prove that there exist distinct indices
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
and
![j](/media/m/7/9/e/79ebb10f98eb80d16b0c785d9d682a72.png)
such that
![a_i + a_j](/media/m/a/d/9/ad99c86f2645376085147080feba4cfa.png)
does not divide any of the numbers
![3a_1](/media/m/d/c/b/dcb4d90342cc02332031f1deb92dd9d3.png)
,
![3a_2](/media/m/b/f/b/bfb7b0c3c26e28fd0efe8fae1544c90c.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![3a_n](/media/m/b/6/6/b665549283349b90f78d1299dbc0f6c4.png)
.
Proposed by Mohsen Jamaali, Iran
%V0
Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive integers, $n\ge 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$, $3a_2$, $\ldots$, $3a_n$.
Proposed by Mohsen Jamaali, Iran