AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
+1 w
be a cyclic quadrilateral. Let , , 
be the feet of the perpendiculars from to the lines , 
, , respectively. Show that 
if and only if the bisectors of 
and 
are concurrent with .
be reals such that 
. Prove that
+1 w
where 
are lengths of parallel sides and 
are other two sides.
be a triangle. A circle passing through and intersects segments and at and , respectively. Lines and 
intersect at 
, while lines 
and 
intersect at 
. Prove that 
if and only if 
.
be real numbers such that 
Prove that
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
then
where 
Also, 
and 
is largest real root of the equation
be a triangle, and let 
and 
be points on segment symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line at . Similarly, let 
denote the circle passing through and tangent to line at 
. Let the circles and intersect again at point 
(
). Prove that 
.
with radii 
are given. is a diameter of 
and the tangent from to 
touches at . What is the length of 
.
and 
Prove that 
and 
.
Prove that
such that 
.
?
then 
Equality holds iff 
and any cyclic permutations.
and 
and 
so by Pythagoras 
.
and 
are similar (Thales) and 
and hence 
and 
.
.