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.
real numbers 
, we can select a real number 
and transform the sequence into 
. This transformation can be performed multiple times, with each chosen real number potentially being different
+1 w
where 
are lengths of parallel sides and 
are other two sides.
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.
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 
.
. On its sides 
and , the points 
and 
are respectively chosen such that the quadrilateral 
is cyclic. Prove that 
. On its sides and , the points and are respectively chosen such that the quadrilateral is cyclic. Prove that 
. On its sides and , the points and are respectively chosen such that the quadrilateral is cyclic. Prove that 
. So we need to prove that
Fix and vary 
along the lines 
such that 
. Then it is known that 
obtains its minimum at the points for which 
is the midpoint of . Thus the LHS is the maximum when is the midpoint of and it suffices to consider this case. be the reflection of 
in . Then it's easy to see that 
and 
. Hence
The last equality holds due to the similarity of the triangles 
and 
.
. On its sides and , the points and are respectively chosen such that the quadrilateral is cyclic. Prove that .
. On its sides and , the points and are respectively chosen such that the quadrilateral is cyclic. Prove that . So we need to prove thatFix and vary along the lines such that . Then it is known that obtains its minimum at the points for which is the midpoint of . Thus the LHS is the maximum when is the midpoint of and it suffices to consider this case. be the reflection of in . Then it's easy to see that and . HenceThe last equality holds due to the similarity of the triangles and .
?
and 
Prove that 
and 
.
Prove that
such that 
.
