MathDB

3

Part of 1988 China National Olympiad

Problems(1)

China Mathematical Olympiad 1988 problem3

Source: China Mathematical Olympiad 1988 problem3

11/5/2013
Given a finite sequence of real numbers a1,a2,…,ana_1,a_2,\dots ,a_n (∗\ast), we call a segment ak,…,ak+l−1a_k,\dots ,a_{k+l-1} of the sequence (∗\ast) a “long”(Chinese dragon) and aka_k “head” of the “long” if the arithmetic mean of ak,…,ak+l−1a_k,\dots ,a_{k+l-1} is greater than 19881988. (especially if a single item am>1988a_m>1988, we still regard ama_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 19881988.
combinatorics unsolvedcombinatorics