IMO 2007 Problem 1

For discussing Olympiad Level Algebra (and Inequality) problems
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Thanic Nur Samin
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IMO 2007 Problem 1

Unread post by Thanic Nur Samin » Wed Aug 24, 2016 10:25 pm

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}$,
\[ \max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (1) \]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (1).
Hammer with tact.

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Zawadx
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Re: IMO 2007 Problem 1

Unread post by Zawadx » Tue Aug 30, 2016 4:52 pm

Hint:
Draw a picture! It'll help you to understand the weird question.

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