Either I did not understand the question, or you don't need Muirhead and it is clear that the asked value is $n-1$??

Valentin Vornicu wrote: Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality \[ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r \] is true for all $r\geq \rho (n)$. All the questions of the test Is it a joke?

ya... is that a joke.. or the ans is not n-1?

It must be n-1. For $r\geq n-1$ it is clearly true by Cebasev+AM-GM and taking the first n-1 numbers equal to x very large we find that the answer must be at least n-1. Thus... it is n-1.

harazi wrote: It must be n-1. For $r\geq n-1$ it is clearly true by Cebasev+AM-GM and taking the first n-1 numbers equal to x very large we find that the answer must be at least n-1. Thus... it is n-1. then......... this question is really too easy for an imo tst..

Just because I managed to solve it? . No, indeed, this is really good for a local olympiad, but not more.

harazi wrote: Just because I managed to solve it? . No, indeed, this is really good for a local olympiad, but not more. No, because I also managed to solve it easily.. The ones you managed to solve r not those i can solve..

harazi wrote: Just because I managed to solve it? . No, indeed, this is really good for a local olympiad, but not more. Indeed. But what can be done about it?

It is not my duty to answer these questions. Yet, I know there were at least two nice and difficult inequalities on the list...

harazi wrote: there were at least two nice and difficult inequalities on the list... let me guess who proposed them