2018 NT exam P2

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Ananya Promi
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2018 NT exam P2

Unread post by Ananya Promi » Sun Apr 15, 2018 10:05 pm

Let $a_1, a_2, ....., a_n$ be a sequence of real numbers such that $a_1+a_2+......a_n=0$ and define $b_i=a_1+a_2+......a_i$ for $$1\leq i \leq n$$. Suppose that $$b_i(a_{j+1}-a_{i+1})\geq 0$$ for all $$1\leq i \leq j \leq n-1$$. Show that
max |$a_l$| $\geq$ max |$b_m$|
$(1 \leq l \leq n)$ and $(1 \leq m \leq n)$

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