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.
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
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
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 the circumcenter of triangle 
and let 
be the height from 
(
). Let 
and 
be the midpoints of 
and 
respectively. Let 
and 
be the circumcircles of triangles 
and 
. Prove that 
.
for 
be a 
square. With some easy cord bash, we find 
so the answer is 
.
be nonnegative real numbers such 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)
is the mid-point of 
.
cuts 
in 
and 
cuts 
point. We can write ratio of areas of 
and 
triangles as 
.
.