AoPS Academy
is written on the board. Every step, we replace the written number 
into the number 
, where 
is a prime divisor of , and this step continues for infinity. Prove that the number 
was chosen as the number infinitely many times.
and 
be side lengths of a triangle with perimeter 
. Show that![\[a^2+b^2+c^2+abc<8.\]](http://latex.artofproblemsolving.com/d/9/8/d98a071538c07301938da20cf2a293b0936d848c.png)
b) Is there a real number 
such that for all triangles with perimeter we have![\[a^2+b^2+c^2+abc<d \quad\]](http://latex.artofproblemsolving.com/c/d/5/cd5e002c21106833f3aa3fb08b67a60d0d5eb10e.png)
where and are the side lengths of the triangle?
be a triangle with incentre 
and circumcircle 
. Let 
and 
be the second intersection points of with 
and 
, respectively. The chord 
meets 
at a point 
, and 
at a point 
. Let 
be the intersection point of the line through parallel to 
and the line through parallel to 
. Suppose that the tangents to at 
and 
meet at a point 
. Prove that the three lines 
and 
are either parallel or concurrent.
be the circumcenter of triangle . The altitudes from 
of triangle intersects the circumcircle of the triangle at 
respectively. 
meets 
at 
respectively. Prove that the circumcircles of triangles 
share two common points.
be an isosceles triangle with 
. Take points 
on side and on side and 
the intersection of bisectors of angles 
and 
such that 
lies on side 
. Prove that is the midpoint of .
we get 
so done.
Then note that
If 
then we must have 
Thus 
Similarly 
Dividing the last two equations, we get 
be the feet of perpendiculars from 
respectively. first 
-
then 
also 
and since 
we have 
so 
so 
of 
,![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
since 
is an isosceles one, therefore 
.
similar to 
similar to 
.
,
,
,
and 
.
relative