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.
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
such that 
.
be a triangle with orthic triangle 
and orthocenter 
and midpoint of 
as 
. If 
then prove that 
passes through pole of wrt 
.
Prove that
with incenter 
,
is a point on 
,the bisector of 
meet 
at 
.Prove that 
be a non-constant function from the set of positive integers into the set of positive integer, such that 
divides 
for all distinct positive integers 
, 
. Prove that there exist infinitely many primes 
such that divides 
for some positive integer 
.
be positive real numbers so that 
. Prove that![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
. Prove that: 
such that 
are four given points on the plane, so that no three of them are collinear. Let 
be the set of conics passing through . Consider a fixed point 
, for each 
, suppose there are distinct points 
, so that 
. Let 
be the line joining the intersection of 
and 
with the intersection of 
and 
. does not depend on the choice of . passes through a fixed point when 
varies."
such that:![\[f(1+xf(y))=yf(x+y)\]](http://latex.artofproblemsolving.com/f/2/a/f2a9e8eb184c921d5f1fa59137d99e38b29c6962.png)
be an acute scalene triangle , be the orthogonal projection of 
on , and 
are the midpoints of 
and 
respectively. Let and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
lies on the side 
of 
. The circumcenters of 
and 
are 
and 
, respectively and 
. The orthocenter of is 
and 
Find the angles of if 
and 
Find the minimum 
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
