[/quote]
,
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
,
,
,
with 
,
,
,
and lines 
and 
intersect at 
.
and 
intersect again at 
.
and 
intersect the circumcircle of triangle 
again at 
and 
,
lies on 
.
for which
(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
.
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
,
where 
is the number of times the number 
appears in the list. So, for example, for the list 
,
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
,
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist 
and 
are acute and 
.
,
.
Y by anantmudgal09, Adventure10, Mango247
Let 
be a scalene triangle with circumcenter 
and incenter 
. The 
-excircle, 
-excircle, and 
-excircle of triangle touch 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the orthocenter of 
and 
be the orthocenter of . Show that if 
is the midpoint of 
, then lines 
and 
are parallel.
Michael Ren
be a scalene triangle with circumcenter 
and incenter 
. The 
-excircle, 
-excircle, and 
-excircle of triangle touch 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the orthocenter of 
and 
be the orthocenter of . Show that if 
is the midpoint of 
, then lines 
and 
are parallel.Michael Ren
be a point on the Darboux cubic of 
with its pedal triangle 
and 
be the perspector of 
If 
are the orthocenters of 
then 
bisects 
be the orthocenter of 
respectively and 
be the isogonal conjugate of 
If 
is the point such that 
then by Pascal's theorem for 
of 
so 
are orthologic and the orthology center of 
WRT 
be the circum-rectangular hyperbola of 
By 
to 
so we conclude that 
be the Bevan point, 
be the intouch triangle and 
be the medial triangle of 
.
so since the orthocenter of 
lies on 
;
