AoPS Academy
the points 
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
. Show that 
. When will equality hold?
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
be the orthocenter of acute triangle , let be the foot of the altitude from to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be a quadrilateral inscribed in circle 
with 
. Let 
and be the reflections of 
over lines 
and , respectively, and let be the intersection of lines 
and 
. Suppose that the circumcircle of 
meets at and 
, and the circumcircle of 
meets at and 
. Show that 
.
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective. is bijective if for every integer 
, there exists exactly one integer such that 
.
with 
, point lies strictly between 
and on side 
, and point lies strictly between and 
on side 
such that 
. The degree measure of 
is 
, where 
and are relatively prime positive integers. Find 
.
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 .
such that 
.
