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.
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
. 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.
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 
.
be a right-angled triangle with hypotenuse . Denote by the foot of the altitude from . Let 
, and be the midpoints of the segments 
, and 
, respectively. Prove that 
.
be a right-angled triangle with hypotenuse . Denote by the foot of the altitude from . Let , and be the midpoints of the segments , and , respectively. Prove that .
and 
Prove that 
and 
.
Prove that
such that 
.
?
then 
Equality holds iff 
and any cyclic permutations.
be the medians of triangle 
Note that triangles 
and 
are similar. Therefore 
and 
Therefore 