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 
.
and 
, do not lie in one plane and intersect at a point . Let 
be center of sphere that passes through the points 
and . Prove that 
and
intersect at one point. (P.Kozhevnikov)
, then the circumcenter is at:![\[\frac{(\mathbf{a\cdot a})(\mathbf{b\times c})+(\mathbf{b\cdot b})(\mathbf{c\times a})+(\mathbf{c\cdot c})(\mathbf{a\times b})}{12V}\]](http://latex.artofproblemsolving.com/5/8/6/5867d30918b2da1262f68b5b88ea601b424e13e7.png)
is the volume.
, 
, and 
, where 
so that is at the origin. Then we have:
in space such that 
for three points 
is the line passing through the circumcenter of 
perpendicular to the plane containing .
passing through their midpoints, which is exactly the line described.
, 
lie on a plane 
. Then by the lemma 
for all 
, so 
for 
. This is true for planes 
and 
so in fact the solid created by all the circumcenters is just a rectangular prism. But we're done, since the space diagonals of a rectangular prism obviously intersect the center.
. Let 
be images of points 
respectively. Let spheres 
intersect by circles 
. To prove that lines 
are concurrent it's enough to prove that circles lie on the same sphere.
of planes 
and 
. Line is image of circle , so it's enough to prove that all lines lie on the same plane.
easy to see that lines , 
intersect at point 
. So any two line , lie on the same plane. So all lines lie on the same plane. 
?
and 
Prove that 
and 
.
Prove that
such that 
.
