Determine the largest natural number m such that for each non negative real numbers a1≥a2≥...≥a2014≥0 , it is true that a1+a2+...+amm≥√a21+a22+...+a220142014
Problem
Source: INAMO Shortlist 2014 A5
Tags: algebra, inequalities
22.05.2019 02:09
Solution to the problem as stated: Clearly for m=2014 the inequality holds. For m=2015 the inequality does not hold, since for any choice of a1,a2,…,a2014, we can make a2015 as small as we want to falsify it.
22.05.2019 07:26
parmenides51 wrote: Determine the largest natural number m such that for each non negative real numbers a1≤a2≤...≤a2014≤0 , it is true that a1+a2+...+amm≥√a21+a22+...+a220142014 Actually the corrected version should be : Determine the largest natural number m such that for each non negative real numbers a1≥a2≥⋯≥a2014≥0, then the following holds. a1+a2+⋯+amm≥√a21+a22+⋯+a220142014
22.05.2019 07:32
I just replaced the incorrect ≤ with the correct ≥, thanks for noticing