be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
with orthocenter 
.
to meet 
at 
.
meets the midpoint of 
.
,
.
.
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
be reals with 
a
.
be non-negative reals such that 
.
people seated around a circular table, and 
cookies
be one of these people. Find the least 
Y by
Let the 
be a quadrilateral without parallel sides, inscribed in a circle. Let 
and 
be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at and are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.
be a quadrilateral without parallel sides, inscribed in a circle. Let 
and 
be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at and are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.This post has been edited 4 times. Last edited by parmenides51, Jun 21, 2022, 1:57 AM
