[/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 
.
be a triangle with circumcircle 
,
.
,
,
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 
.
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
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 Adventure10, Mango247
In 
, we have 
. 
are the tangent points of the inscribed circle of with the line segments 
respectively. The points 
are the midpoints of the line segments 
. The straight line 
intersects with ray 
at 
, straight line 
intersects ray 
at 
and the straight line 
intersects ray 
at 
. Prove that 
.
, we have 
. 
are the tangent points of the inscribed circle of with the line segments 
respectively. The points 
are the midpoints of the line segments 
. The straight line 
intersects with ray 
at 
, straight line 
intersects ray 
at 
and the straight line 
intersects ray 
at 
. Prove that 
.
lie on the radical axis of the circumcircle and incircle of 
