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 
.
,
.
with side length of 
.
is selected on the diagonal 
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 Adventure10, Mango247
Three circles with the same radius 
all pass through point 
(fig. ). Prove that the circle passing through the points where pairs of circles intersect (that is, points
,and 
) also has the same radius .
all pass through point 
(fig. ). Prove that the circle passing through the points where pairs of circles intersect (that is, points
,and 
) also has the same radius .
This post has been edited 2 times. Last edited by parmenides51, Aug 21, 2019, 1:02 AM
