which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
,
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
Y by mahanmath, Adventure10, Mango247
We have connected four metal pieces to each other such that they have formed a tetragon in space and also the angle between two connected metal pieces can vary.
In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane.
proposed by Erfan Salavati
In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane.
proposed by Erfan Salavati
This post has been edited 1 time. Last edited by goodar2006, Sep 13, 2011, 6:54 PM
= Equal Ratios In Quaqdrilateral 
.
Call points 
on 
respectively special if they are coplanar for some arbitrary orientation of the tetragon so that it's not on one plane. Let 
be the signed distances of 
to the plane. We have
lie on 
so that 
are coplanar, and define 
as the distances from 
,
Hence we've shown that 
