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
Two lines 
and 
intersect in a unique point 
. A point 
moves along with constant speed, while another point 
moves along with the same speed. They both pass through the point , but not at the same time.
Show that there is a fixed point
such that the points 
and lie on a common circle all the time.
and 
intersect in a unique point 
. A point 
moves along with constant speed, while another point 
moves along with the same speed. They both pass through the point , but not at the same time.Show that there is a fixed point
such that the points 
and lie on a common circle all the time.
be the angle bisector of 
starting from 
along a line 
which is parallel to 
be the reflection of 
be a point that also moves along 
are at the same moment in 
.
intersects 
