and for 
,
Prove that every natural number occurs exactly once in this sequence.
inscribed in 
,
is the midpoint of arc 
of 
intersects 
at 
.
intersects 
at 
different from 
is the orthocenter of triangle 
= 
for which the number![\[x^4 + x^3 + x^2 + x + 1\]](http://latex.artofproblemsolving.com/a/f/f/affe758628a67ad6f72e82decfe71356c24daa6e.png)
?
Y by Adventure10, Mango247
Consider a circle 
and a point 
outside 
A line 
different from 
cuts 
at 
and 
where lies between and 
Now the symmetric line of with respect to axis of symmetry the line cuts at 
and 
where lies between and 
Show that the diagonals of the quadrilateral
intersect in a fixed point.
and a point 
outside 
A line 
different from 
cuts 
at 
and 
where lies between and 
Now the symmetric line of
with respect to axis of symmetry the line cuts at 
and 
where lies between and 
Show that the diagonals of the quadrilateral
intersect in a fixed point.
,
- contact point of a tangent to 
;
=
,
,
is cyclic and 
,
fixed.
