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 nonnegative real numbers such that 
. Find min
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
denote the sum of the sum of the factors of all positive 
, for example, 
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
, which would be 
. Let 
denote 
. Show that 
or find a counterexample
such that
be nonnegative real numbers such that 
Prove that
Let be nonnegative real numbers such that 
Prove that
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
such that the numbers![\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]](http://latex.artofproblemsolving.com/b/3/0/b303b6dd25b8e172f41737ffa6c136dcc4db2266.png)
are distinct modulo 
for all integers 
with 
.
is defined by 
and 
for 
.)
such that for every prime number 
and natural number 
,
is a complete residue system modulo . With 
for every natural number 
and 
.
be positive real numbers such that 
. Prove that![\[ \frac{1}{a+2b}+\frac{1}{b+2c}+\frac{1}{c+2a} \le 1\]](http://latex.artofproblemsolving.com/d/1/4/d14ce64b53e3e4ebd49cddc07c6fa7bb4257311a.png)
be positive real numbers. Prove that
![\[ \overset{n}{\underset{i=1}{\sum}} \frac{1}{a_i+n-1} \le 1\]](http://latex.artofproblemsolving.com/d/6/b/d6b71e62d8728548b4dce109e9dac1a9b5e29b77.png)
are positive real numbers such that 
.
and 
Prove that
Prove that
,![\[abc\sqrt{1-d}+bcd\sqrt{1-a}+cda\sqrt{1-b}+dab\sqrt{1-c}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/8/698270a4b6b055b71f9191de370eb674f1ff72f5.png)
