[/quote]
,
be a triangle with circumcircle 
,
,
,
with 
,
,
,
intersect at 
and lines 
and 
intersect at 
.
and 
intersect again at 
.
and 
intersect the circumcircle of triangle 
again at 
and 
,
lies on 
.
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 
.
for which
(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
.
and 
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 
side length and with center 
.
is drawn where point 
.
,
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
Let 
be a cyclic quadrilateral. Circle with diameter 
intersects 
, 
, 
, and 
in 
, 
, 
, and 
, respectively (all different from 
and 
). The lines 
and 
intersect in 
. Prove that the bisector of 
and the line 
are perpendicular.
be a cyclic quadrilateral. Circle with diameter 
intersects 
, 
, 
, and 
in 
, 
, 
, and 
, respectively (all different from 
and 
). The lines 
and 
intersect in 
. Prove that the bisector of 
and the line 
are perpendicular.
where, 
are parallel & 
are non-parallel, then, if 
,
,
are isosceles triangles 
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/a/a/1/aa17a93b9251df5710980fd934ed1b5919f458eb.png)
,
is an isosceles trapezium, hence by a corollary of our 
and since, 
solves the problem
