IMO Shortlist 1991 problem 24
Dodao/la:
arhiva2. travnja 2012. Suppose that
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
and
![x_1, x_2, \ldots, x_n](/media/m/0/6/a/06a7dfb24aaec0d40a6067e2b1a7d750.png)
are real numbers between 0 and 1 (inclusive). Prove that for some index
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
between
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
and
![n - 1](/media/m/b/9/f/b9f2e24ffd917df5f63d30599dd3220c.png)
the
inequality
%V0
Suppose that $n \geq 2$ and $x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $i$ between $1$ and $n - 1$ the
inequality
$$x_i (1 - x_{i+1}) \geq \frac{1}{4} x_1 (1 - x_{n})$$
Izvor: Međunarodna matematička olimpijada, shortlist 1991