IMO Shortlist 1983 problem 6
Dodao/la:
arhiva2. travnja 2012. Suppose that
![{x_1, x_2, \dots , x_n}](/media/m/6/d/9/6d9ec0addee65eea10f32971de53e382.png)
are positive integers for which
![x_1 + x_2 + \cdots+ x_n = 2(n + 1)](/media/m/f/e/e/feec81af81d94d83b7a0944e466d3d72.png)
. Show that there exists an integer
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
with
![0 \leq r \leq n - 1](/media/m/1/a/e/1aee04c67657e1eae66a4fd9b378e1d0.png)
for which the following
![n - 1](/media/m/b/9/f/b9f2e24ffd917df5f63d30599dd3220c.png)
inequalities hold:
![x_{r+1} + \cdots + xn + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.](/media/m/0/7/6/0762f56734e7378aaa8ae333bf2f119d.png)
Prove that if all the inequalities are strict, then
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
is unique and that otherwise there are exactly two such
%V0
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
$$x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r;$$
$$x_{r+1} + \cdots + xn + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.$$
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$
Izvor: Međunarodna matematička olimpijada, shortlist 1983