[/quote]
,
(use in a problem for junior) to practice. Can anyone share to me?
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
,
.
,
,
with 
,
,
,
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
be an acute angled scalene triangle with circumcircle 
and 
be the circumcenter and orthocenter of 
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 
.
,
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 
,
Y by ItsBesi
Let 
be a triangle with 
. Let 
be a point on side 
such that 
. Consider the circle 
passing through point and tangent to side 
at point 
. Consider the circumscribed circle 
of the triangle that interesects the circle at points and 
. Prove that point is the intersection point of the perpendicular bisectors of line segments 
and 
.
be a triangle with 
. Let 
be a point on side 
such that 
. Consider the circle 
passing through point and tangent to side 
at point 
. Consider the circumscribed circle 
of the triangle that interesects the circle at points and 
. Prove that point is the intersection point of the perpendicular bisectors of line segments 
and 
.
intersect circle 
.
= 
= 
.
is an isosceles trapeziod, with 
,
= 
is isosceles. Hence triangle 
is also isosceles
?
