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.
: 
such that:
write all its divisors in increasing order: 
. such that 
.
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
, and circles of diameters 
and 
intersect at points 
and 
. The circles of diameters 
and 
intersect in two points 
and 
. Prove that the lines 
and 
are concurrent.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
for which there exists a sequence 
of positive integers satisfying the following conditions:[list]
for all integers 
for all integers 
the sequence 
is a permutation of
[/list]
.
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
. Let the lines 
and 
meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
be non-negative real numbers such that 
Prove that
be positive real numbers such that 
and 
, prove that
and 
outside triangle 
such that 
and 
. Connect 
and 
, which intersect at point 
. Let 
intersect 
at point . Prove that 
.
lies on the circumscribed circle around 
, and are projections of point on and respectively. Prove that, if 
and 
are the middle points of and respectively, then 
is perpendicular to 
.
, 
. By Gliding principle 
is also similar to them. So 
.
is projection of point 
on 
and 
. 
and colinear, by Simson line.
-cyclic. 
, 
and 
, 
are median of 
, 
, respectively.
, so 
-cyclic. Therefore, 
lies on the circumscribed circle around , and are projections of point on and respectively. Prove that, if and are the middle points of and respectively, then is perpendicular to .
