[/quote]
,
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 
then prove that 
passes through pole of 
.
Prove that
,
meet 
at 
.
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 
,
.
such that 
for some positive integer 
.
be positive real numbers so 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 
and 
Find the minimum 
are four given points on the plane, so that no three of them are collinear. Let 
be the set of conics passing through 
,
,
,
.
be the line joining the intersection of 
and 
with the intersection of 
and 
.
varies."
such that:![\[f(1+xf(y))=yf(x+y)\]](http://latex.artofproblemsolving.com/f/2/a/f2a9e8eb184c921d5f1fa59137d99e38b29c6962.png)
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 
on 
respectively. Let 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
Y by
In triangle 
, the inscribed circle 
is tangent to sides 
, 
, 
at points 
respectively. Let 
and 
be the midpoints of 
and 
respectively. Points 
are considered on the line 
such that 
, 
and 
. Show that the lines 
, 
and 
are concurrent.
, the inscribed circle 
is tangent to sides 
, 
, 
at points 
respectively. Let 
and 
be the midpoints of 
and 
respectively. Points 
are considered on the line 
such that 
, 
and 
. Show that the lines 
, 
and 
are concurrent.
