Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer and let
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
,
![a_2](/media/m/4/0/1/401f4cdfec59fba73ae32fa6769c72cb.png)
,
![a_3](/media/m/e/5/1/e517d36771b6a4db32de5ee281210809.png)
, ...,
![a_k](/media/m/8/f/f/8ffe60c23d3334cc61d0660473bf1b61.png)
(
![k \geqslant 2](/media/m/a/c/2/ac29cfb6e283b999e6e17ec8e26b7345.png)
) be distinct integers in the set
![\left\{1,\,2,\,\ldots,\,n\right\}](/media/m/e/d/9/ed9c740a1b23c7e3d90350414f9d0f50.png)
such that
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
divides
![a_i \left(a_{i + 1} - 1\right)](/media/m/5/1/c/51ce1a8b99ae24d58e6475c9f2a5242f.png)
for
![i = 1,\,2,\,\ldots,\,k - 1](/media/m/9/1/e/91e4ff0bd638adab3140c658ed205817.png)
. Prove that
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
does not divide
![a_k \left(a_1 - 1\right)](/media/m/8/c/e/8ce3c5d3c917cd35cd523d16f08d3374.png)
.
Proposed by Ross Atkins, Australia
%V0
Let $n$ be a positive integer and let $a_1$, $a_2$, $a_3$, ..., $a_k$ ($k \geqslant 2$) be distinct integers in the set $\left\{1,\,2,\,\ldots,\,n\right\}$ such that $n$ divides $a_i \left(a_{i + 1} - 1\right)$ for $i = 1,\,2,\,\ldots,\,k - 1$. Prove that $n$ does not divide $a_k \left(a_1 - 1\right)$.
Proposed by Ross Atkins, Australia