Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds \[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]
Problem
Source: Romanian IMO Team Selection Test TST 1999, problem 4
Tags: inequalities, search, inequalities unsolved, Hi
harazi
24.02.2005 21:42
Romania 1999 TST and it has been posted at least 6 times! Please, search it in solved section.
Celeborn
24.02.2005 21:45
Oops...thank you anyway! (the worksheet I got this from said it was Vietnam 98!)
chinacai
10.08.2010 13:06
who can solve the question？
arqady
10.08.2010 16:36
chinacai wrote: who can solve the question？ See here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1965995#p1965995
colorfuldreams
10.08.2010 16:40
arqady wrote: chinacai wrote: who can solve the question？ See here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1965995#p1965995 can it solve by Jensen ?
arqady
10.08.2010 16:52
colorfuldreams wrote: arqady wrote: chinacai wrote: who can solve the question？ See here: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1965995#p1965995 can it solve by Jensen ? Certainly, by the similar way: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1967083#p1967083
sqing
16.12.2016 16:38
http://www.artofproblemsolving.com/community/c6h1101773p4968742 Let $x_i$ be positive real numbers such that $\sum _{i=1}^n \frac{1}{x_i} =\sum _{i=1}^n x_i $ , show that: $$\sum _{i=1}^n \frac{1}{n-1+x_i} \le 1$$here
Rhapsodies_pro
01.03.2022 13:16
chinacai wrote: who can solve the question？ See also https://artofproblemsolving.com/community/h2050333.