IMO Shortlist 2001 problem C1
Dodao/la:
arhiva2. travnja 2012. Let
![A = (a_1, a_2, \ldots, a_{2001})](/media/m/d/e/a/dea1bb6c045d836086dac9a2662ba5c6.png)
be a sequence of positive integers. Let
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
be the number of 3-element subsequences
![(a_i,a_j,a_k)](/media/m/e/2/c/e2c5aee66748fb5f5ae6f6de58c3e546.png)
with
![1 \leq i < j < k \leq 2001](/media/m/7/8/a/78aee77a7aa45f72d01c1e0c4f1829a9.png)
, such that
![a_j = a_i + 1](/media/m/8/d/b/8db3887b5379b05667605bcab808066f.png)
and
![a_k = a_j + 1](/media/m/5/8/5/585dc667a841b5401156af6eebd83025.png)
. Considering all such sequences
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, find the greatest value of
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
.
%V0
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
Izvor: Međunarodna matematička olimpijada, shortlist 2001