AoPS Academy
, 
, 
are pairwise distinct real numbers satisfying 
Prove that
Let , , are pairwise distinct real numbers satisfying 
Prove that
and 
be positive integers. is a divisor of 
and is a divisor of 
.
.
and 
be the two foci of ellipse 
. 
is a point on . Let 
be the center of the excircle of 
. When 
, find the minimum eccentricity of .
of natural numbers with 
elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).
and 
touch a circle at 
and 
, respectively, and meet perpendicularly at 
. 
is on , is on , and 
is on the circle, so that 
is a rectangle with 
and 
. Determine the radius of the circle.
and touch a circle at and , respectively, and meet perpendicularly at . is on , is on , and is on the circle, so that is a rectangle with and . Determine the radius of the circle.
is a rectangle because it has 4 right angles and since 
we see that 
.
because 
is a rectangle. We have that 
.
from the rectangle 
.
.
since 
and 
we must have that 
.
solving we get 
is either 
and since 
we know that 