[/quote]
,
be an acute triangle with 
.
intersects 
at 
,
at 
and 
at 
respectively. Suppose 
and 
intersect again at 
,
is the angle bisector of 
.
and 
Prove that
Let 
Prove that
be complex numbers satisfying 
.
complex numbers such that the modulus of the sum of these complex numbers is not less than 
.
and 
be three positive integers. Prove that there exist a positive integer 
and a set of 
positive integers 
,
, the 
-
-
-
.
, s.t :
be an acute triangle with 
,
of side 
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
satisfy,
Where 
}
has a unique solution in integers with 
and 
.
as 
fir 
.
for all 
.
and orthocenter 
.
intersect 
respectively. Lines through 
intersects 
at 
is a parallelogram. Prove that lines 
and 
.
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
are positive real numbers.Prove that
Y by Funcshun840, PikaPika999
Francisco wrote a sequence of numbers starting with 
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
with
and 
and 
