IMO Shortlist 2007 problem A1
Dodao/la:
arhiva2. travnja 2012. Real numbers
![a_{1}](/media/m/0/6/5/0653090dabb5d1972cd7a7dfcd31abc1.png)
,
![a_{2}](/media/m/5/5/6/5565dac5c7f1dadb0e60c273c1d11c80.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![a_{n}](/media/m/e/1/b/e1bf963ddae5d084fba54d8a7aa04acc.png)
are given. For each
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
,
![(1 \leq i \leq n )](/media/m/3/4/2/3422ede776c0c7a04aafb35d35756c12.png)
, define
and let
![d = \max \{d_{i}\mid 1 \leq i \leq n \}](/media/m/d/f/f/dffec58d3ba7280f1d6be8191c74a9bc.png)
.
(a) Prove that, for any real numbers
![x_{1}\leq x_{2}\leq \cdots \leq x_{n}](/media/m/c/3/6/c3665832c4e72b2c7fd7c991e8364036.png)
,
(b) Show that there are real numbers
![x_{1}\leq x_{2}\leq \cdots \leq x_{n}](/media/m/c/3/6/c3665832c4e72b2c7fd7c991e8364036.png)
such that the equality holds in (*).
Author: Michael Albert, New Zealand
%V0
Real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ are given. For each $i$, $(1 \leq i \leq n )$, define
$$d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$$
and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
$$\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$$
(b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
Author: Michael Albert, New Zealand
Izvor: Međunarodna matematička olimpijada, shortlist 2007