MathDB

48

Part of 1982 IMO Longlists

Problems(1)

There exists a finite sequence a_i for all sequences c_i

Source: IMO LongList 1982 - P48

5/16/2011
Given a finite sequence of complex numbers c1,c2,…,cnc_1, c_2, \ldots , c_n, show that there exists an integer kk (1≤k≤n1 \leq k \leq n) such that for every finite sequence a1,a2,…,ana_1, a_2, \ldots, a_n of real numbers with 1≥a1≥a2≥⋯≥an≥01 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0, the following inequality holds: ∣∑m=1namcm∣≤∣∑m=1kcm∣.\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.
inequalitiesinductioncomplex numberscombinatorial geometryalgebra unsolvedalgebra