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 the set of all nonnegative integers. Find all the functions 
satisfying the relation![\[ f(f(f(n))) = f(n+1 ) +1 \]](http://latex.artofproblemsolving.com/8/b/9/8b953428105df2c2344963cbaf3eb81d8201e0df.png)
.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
, such that for any 
, 
very convex if![\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|
\]](http://latex.artofproblemsolving.com/3/b/a/3ba7733caf51cc70a090920ca21f8b47aad9b1ec.png)
and 
. Prove that no very convex function exists.
satisfying
for all integers 
.
, and for any positive integer 
, 
means applied times to , and 
means 
applied times to .
satisfying 
.Prove that:
such that 
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers .
be an isosceles trapezoid with 
. Let 
be the midpoint of 
. Denote by and the circumcircles of the triangles 
and 
, respectively. Let be the crossing point of the tangent to at with the tangent to at . Prove that 
is tangent to .
consecutive positive integers, none of which could be written as 
for some integers 
? Justify your answer.
that 
.
will be odd.
then,
.
and 
.
.
, a contradiction. and is prime number that then 
which implies that 
and so 
.
If positive integers 
that 
when is prime number that then can't be written in the form 
. such that
when 
is different prime number such that 
. that satisfies all of the above conditions. 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
