Let x,y,z and a,b,c be positive real numbers with a+b+c=1. Prove that (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2)≥19.
Problem
Source: Mediterranean Mathematics Olympiad 2017, Problem 4
Tags: inequalities
15.06.2017 14:54
Power mean ⟹(a32+b32+c323)2≥(a+b+c3)3=127⟹(a32+b32+c32)2≥13 C-S(T2) ⟹(a3x2+2y2+b3y2+2z2+c3z2+2x2)≥(a32+b32+c32)23(x2+y2+z2)≥19(x2+y2+z2) (P.S.Please check,I have did somethink silly yesterday )
15.06.2017 16:52
15.06.2017 17:35
socrates wrote: Let x,y,z and a,b,c be positive real numbers with a+b+c=1. Prove that (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2) ≥ 19 Using Holder we have (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2)≥(x2+y2+z2)(19(x2+y2+z2))=19.
19.06.2017 06:14
socrates wrote: Let x,y,z and a,b,c be positive real numbers with a+b+c=1. Prove that (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2) ≥ 19 2017 Mediterranean Mathematics Olympiad . Problem 4 Proof of Zhangyunhua: (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2) =13((x2+2y2)+(y2+2z2)+(z2+2x2))(a3x2+2y2+b3y2+2z2+c3z2+2x2) ≥13(x32+y32+z32)2≥19
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19.11.2018 16:11
sqing wrote: socrates wrote: Let x,y,z and a,b,c be positive real numbers with a+b+c=1. Prove that (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2) ≥ 19 2017 Mediterranean Mathematics Olympiad . Problem 4 Proof of Zhangyunhua: (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2) =13((x2+2y2)+(y2+2z2)+(z2+2x2))(a3x2+2y2+b3y2+2z2+c3z2+2x2) ≥13(x32+y32+z32)2≥19 ≥13(x32+y32+z32)2≥19 can you explain this part pls? thanks
22.11.2018 22:25
AlgebraFC wrote:
what is holder's inequality?
23.11.2018 01:26
You can use Google search, but anyway see here to learn many classical inequalities (Holder's inequality is also here) : https://brilliant.org/wiki/classical-inequalities/
29.09.2021 21:43
Absolutely trivial. By Holder, we have ((x2+2y2)+(y2+2z2)+(z2+2x2))(1+1+1)(a3x2+2y2+b3y2+2z2+c3z2+2x2)≥(a+b+c)3.Rearrange and we get desired.
29.09.2021 22:12
Exactly same as above, and I also think this is trivial. By Holder we have (∑cyca3x2+2y2)⋅(∑cycx2+2y2)⋅(∑cyc1)≥1. Rearranging this gives the result.
29.08.2023 00:12
Notice that ∑cycx2⋅∑cyca3x2+2y2Generalized T2'S≥∑cycx2⋅(∑cyca)39∑cycx2=∑cycx29∑cycx2=19◼.