Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ($\ast$), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ($\ast$) a “long”(Chinese dragon) and $a_k$ “head” of the “long” if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$. (especially if a single item $a_m>1988$, we still regard $a_m$ as a “long”). Suppose that there is at least one “long” among the sequence ($\ast$), show that the arithmetic mean of all those items of sequence ($\ast$) that could be “head” of a certain “long” individually is greater than $1988$.
Problem
Source: China Mathematical Olympiad 1988 problem3
Tags: combinatorics unsolved, combinatorics
ociretsih
25.12.2014 17:16
Given a "long" $a_k,...,a_{k+l-1}$ suppose also $a_{k+1},...,a_{k+l-1}$,...,$a_{k+s},...,a_{k+l-1}$ are also "longs", but $a_{k+s+1},...,a_{k+l-1}$ is not a "long", then the mean value of the "heads" $a_k,a_{k+1},...,a_{k+s}$ is greater than 1988. Just continue grouping "heads" in this way until all "heads" are taken account of.
Mathzoo
03.03.2016 05:29
Sorry for reviving, but can anyone clarify this? ociretsih wrote: Given a "long" $a_k,...,a_{k+l-1}$ suppose also $a_{k+1},...,a_{k+l-1}$,...,$a_{k+s},...,a_{k+l-1}$ are also "longs", but $a_{k+s+1},...,a_{k+l-1}$ is not a "long", then the mean value of the "heads" $a_k,a_{k+1},...,a_{k+s}$ is greater than 1988 I mean that I have not fully understood how he reached that conclusion.
Mathzoo
03.03.2016 16:09
Bump!.......
Mathzoo
05.03.2016 03:41
Hallo! Any answers to my confusion regarding the solutions?
skyletter
07.03.2016 13:17
I will prove this by induction.(maybe not the easiest way) For $n=1$ is normal.Suppose the statement is true for the number of terms less than $n$. Consider the sequence $\ast$.If $a_1$ is not head,then by induction hypothesis,the conclusion follows.Assume $a_1$ is head of a long,and $a_1,a_2,……,a_q$ is the long with head $a_1$.The average of these $q$ numbers is great than $1988$.Note that if a term in $\ast$ is not a head,it is no greater than $1988$ (else itself could be a long individually).I delete them,the average stays to be greater than $1988$.And for the sequence $a_{q+1},……,a_n$ has two conditions,$1$.no head in it,$2$.has the same head as the part of this in $\ast$.By induction hypothesis,the average of heads in it is greater than $1988$.Done