AoPS Academy
, are there infinitely many positive integers 
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
the set 
can be partitioned into 
triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
be a triangle with circumcircle 
, circumcenter 
, and orthocenter 
. Assume that 
and that 
. Let 
and 
be the midpoints of sides 
and 
, respectively, and let 
and 
be the feet of the altitudes from 
and 
in 
, respectively. Let 
be the intersection of line 
with the tangent line to at 
. Let 
be the intersection point, other than , of with the circumcircle of 
. Let 
be the intersection of lines 
and 
. Prove that 
.
be the foot of perpendicular from to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle . A circle 
with centre 
passes through and , and it intersects sides and at 
and 
respectively. Let be the foot of altitude from to 
, and let be the midpoint of . Prove that the circumcentre of triangle 
is equidistant from and .
is determined by 
and the following recurrence relation for 
:![\[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\]](http://latex.artofproblemsolving.com/f/e/5/fe584783ca31a8ddf8cb2d7e1d18d110e3eaa406.png)
as 
given by ![$f(x) = \sqrt[3]{x + \dfrac {15} {64}}$](http://latex.artofproblemsolving.com/5/7/a/57a844d7180dd2ddcdca6ce73775ea8fbd6f3a97.png)
. It has one real root at 
, it is increasing, convex on ![$(-\infty,r]$](http://latex.artofproblemsolving.com/d/7/7/d7725672d062e8dbf42ca839cfbb73b349d4ad9d.png)
and concave on 
. The equation 
has three real roots at 
, 
and 
.
. The behaviour of as 
depends on the starting term , according to well-known techniques that resume to just considering the graph of 
and its intersection with the main bisector 
, and will be described by:
for 
; 
for 
; 
for 
.
