[/quote]
,
where 
is a positive integer and 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
such that 
divides 
for all positive integers 
and 
with 
.
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
,
be the intouch triangle with 
,
,
,
be the intersecttion of the perpendicular from 
with 
.
and 
the set 
can be partitioned into 
,
,
.
and that 
.
and 
be the midpoints of sides 
be the feet of the altitudes from 
and 
in 
,
with the tangent line to 
.
.
be the intersection of lines 
and 
.
with centre 
passes through 
is equidistant from 
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
,
,
,
satisfy 
and 
.
of 
such that 
.
and 
)
Prove that if lines 
are similar.
such that 
, where 
are non-negative reals such that 
.
.
Y by Adventure10, Mango247
It is given triangle 
and points 
and 
on sides 
and 
, respectively, such that 
. Let 
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
, and 
and 
intersection points of lines 
and 
with side 
. If 
, prove that circle contains intersection point of angle bisector of 
with
and points 
and 
on sides 
and 
, respectively, such that 
. Let 
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
, and 
and 
intersection points of lines 
and 
with side 
. If 
, prove that circle contains intersection point of angle bisector of 
with 
