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 
.
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 
be one of these people. Find the least 
Y by zone5598, Adventure10, Mango247
Rays 
and 
forming an angle of 
are drawn from the same point. Let 
be a fixed point on . For each circle 
tangent to at and intersecting at 
and 
, let 
be the intersection point of the bisector of angle 
with . Describe the locus of and justify your answer.
and 
forming an angle of 
are drawn from the same point. Let 
be a fixed point on . For each circle 
tangent to at and intersecting at 
and 
, let 
be the intersection point of the bisector of angle 
with . Describe the locus of and justify your answer.
,
such that 
.
is tangent to 
.
.
is isosceles, which means that 
.
.
,
,
.
,
lies in the angle bisector of 
,
.
,
(without segment 
itself).
