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 
meets the midpoint of 
,
.
.
is selected on the diagonal 
.
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
Y by Adventure10
A circle 
is given with a non-diameter chord 
. On the tangent at point 
select point 
and mark 
the intersection of the circle with the interior of the line 
(if any). Let 
a point in circle such that quadrilateral 
is a trapezoid . Determine the set of intersections of lines 
and 
belonging to all such trapezoids.
is given with a non-diameter chord 
. On the tangent at point 
select point 
and mark 
the intersection of the circle with the interior of the line 
(if any). Let 
a point in circle such that quadrilateral 
is a trapezoid . Determine the set of intersections of lines 
and 
belonging to all such trapezoids.This post has been edited 1 time. Last edited by parmenides51, Jan 31, 2020, 11:24 AM
