be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
,
,
,
in the plane, will mark the isogonal conjugate of 
,
such that lines 
and 
are reflections around the bisector of 
,
and 
are reflections around the bisector of 
,
and 
are reflections around the bisector of 
.
be reals such that 
and 
Prove that
Where 
be real numbers. Find the ranges of 
converges.
are group. If 
,
or 
.
has at most one solution continuous for 
are complex numbers such that
and if 
denotes the sum of all products of these 
numbers taken 
at a time, prove that
whenever the denominator of the left-hand side is different from 
.
for any positive reals satifying 
?
are real number satisfying
show that the equation 
has at least one real root.
converge?
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I often request that posters use the search function before posting a [not so] new problem (look at the little magnifying glass symbol just at the right of the forum name on the top of forum window, or also here).
I received in PM some remarks about the fact that search function seems very often to give tons of results without any link with the searched problem. Here are some tips I frequently use :
1) use quotes for strings : search "P(x^2-1)" instead of P(x^2-1)
2) dont hesitate to use the character "+" which means "mandatory"
3) Split long strings in order to find problems where strings are not in the same order :
Instead of +"f(2x+f(y))=f(2x)+f(f(y))+xf(2y)", prefer +"f(2x+f(y))" +"f(2x)" +"f(f(y))" +"xf(2y)"
So you'll find
4) If the problem you are looking for contains strings as "2015" or "2016", try to exclude these strings from search term since this is likely an "annual problem" and it may have been posted wu=ith older years?
For example +"f(x-f(y)" +"f(x+y^" +"f(f(y)+y^" will find
as well as 
5) if the problem you are looking for uses function
or 
, dont hesitate to search also for function name 
which is generally the name of the "unknown" function in a FE problem.
6) Dont hesitate also to search for similar problem, just swapping variables
(for example search also for FE "
" when you want to solve "
")
For example :
Searching for problem
:
+"P(x^2-1)" +"P(-x)"
(I avoid using +"P(x)P(-x)" which would miss
or 
)
Searching for problem
:
+"f(x+yf(x))" +"f(f(x))"
(I avoid including the +f(y) in the second item in order to find both f(y)+f(f(x)) and f(f(x))+f(y))
Searching for problem
:
+"f(x+y)+f(x)f(y)" +"f(xy)+f(x)+f(y)"
(I prefered using
instead of $g())
I received in PM some remarks about the fact that search function seems very often to give tons of results without any link with the searched problem. Here are some tips I frequently use :
1) use quotes for strings : search "P(x^2-1)" instead of P(x^2-1)
2) dont hesitate to use the character "+" which means "mandatory"
3) Split long strings in order to find problems where strings are not in the same order :
Instead of +"f(2x+f(y))=f(2x)+f(f(y))+xf(2y)", prefer +"f(2x+f(y))" +"f(2x)" +"f(f(y))" +"xf(2y)"
So you'll find
4) If the problem you are looking for contains strings as "2015" or "2016", try to exclude these strings from search term since this is likely an "annual problem" and it may have been posted wu=ith older years?
For example +"f(x-f(y)" +"f(x+y^" +"f(f(y)+y^" will find
as well as 
5) if the problem you are looking for uses function
or 
, dont hesitate to search also for function name 
which is generally the name of the "unknown" function in a FE problem.6) Dont hesitate also to search for similar problem, just swapping variables
(for example search also for FE "
" when you want to solve "
")For example :
Searching for problem
:+"P(x^2-1)" +"P(-x)"
(I avoid using +"P(x)P(-x)" which would miss
or 
)Searching for problem
:+"f(x+yf(x))" +"f(f(x))"
(I avoid including the +f(y) in the second item in order to find both f(y)+f(f(x)) and f(f(x))+f(y))
Searching for problem
:+"f(x+y)+f(x)f(y)" +"f(xy)+f(x)+f(y)"
(I prefered using
instead of $g())This post has been edited 7 times. Last edited by pco, Jan 26, 2021, 9:36 AM
Reason: Added tip 6
Reason: Added tip 6
