be positive reel numbers; Prove that:

Solution: assuming that,   we get ,

    then by Chebychev,

and by Cauchy-Scwarz tow times,

then

  be a positive reel numbers such as,   . Prove that;

Ukraine, 2001

Solution: we can rewrite the inequality as:

or again ,

assume that                and      ,

we get that    then by Chebychev,

     (*)

and we have that,  

and     then,

from (*) and (**) we deduct that,

  such as  , find the maximum of:

                                           

Solution: first we remarque that,

put that,   and    therefore

we have           

thus,  

then  equality holds when ,

  has at least one reel root, prove that ,  

Tournament of the Towns, 1993

Solution: let  be a root of the equation, therefore, by Cauchy-Schwarz,

   and we have that,

then    

, Prove that:

Gazeta matematicã

Solution:

assume that,     therefore by Chebychev,

Be Cauchy-Schwarz,    

and also   , then

    (*)

By AM-GM,  

and  then  (**)

After summation of (*) and (**) we get,

Junior TST 2002,Romania

Solution: Put that, , therefore, by AM-GM

 and

and we have,         if       

then                

wich implies that,

 , prove that:

Solution: Be Cauchy-Schwarz

   then;

    and  

after summation we get;

   and  are all integers,

Solution:

Multiplying all equations we get

is an integer.we have where is an integer
suppose that one of is equal to for example then,
is integer then is integer wich implies (*)
is integer then is integer wich implies and from wich is false. therefore thus
therefore or
if then
wich is false, thus,


and we have therefore,


suppose that therefore,
if then,
wich don’t have any integer solution.
the or
if


then
if


then
finally the only solution for the problem is , with all permutation possible.

, such as  prove that:

Solution:  let put and and
then the inequality becomes;
with .
We have because
then

By Shur,
then by AM-GM therefore: