AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be reals such that 
Show that
Let 
and Show that
+1 w
. Two subsets 
and 
of 
are said to be friends if the following conditions are true: equals the product of all elements from . that are friends such that each one of them contains at least 
elements.[/quote]
be positive real numbers. Prove that ![\[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]](http://latex.artofproblemsolving.com/0/2/7/027279b75a2b5b2bb71200d315ff4a77b7701aa3.png)
be reals such that 
Show that
Let and Show that
distinct divisors of a positive integer number 
form an arithmetic progression. Prove that the number has at least 
divisors.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
positive integers. Product of any one of them with sum of remaining numbers increased by 
is divisible with sum of all numbers. Prove that sum of squares of all numbers is divisible with sum of all numbers
positive integers. Product of any one of them with sum of remaining numbers increased by is divisible with sum of all numbers. Prove that sum of squares of all numbers is divisible with sum of all numbers
and 
.
satisfy that: 
.
and 
Find the value range of 
so S devides 
and now it's trivially
,
so 
for all 
.
q.e.d.