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.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
such that 
that satisfy the equation
with incenter 
. The incircle of triangle is tangent to 
at 
. Let 
and 
be points on the side BC such that 
and 
, respectively. Let 
and 
be the incenter of triangles 
and 
, respectively. Prove that 
is the Euler line of triangle 
.
, , 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of and , the point 
is the reflection of midpoint of 
. Also let 
be the reflection of 
midpoint of 
. Show that 
concyclic.
. Prove that: 
be a triangle with orthocenter 
and midpoints 
, and 
of , 
, and 
respectively. A circle with center intersects the lines 
, 
, and 
in points 
respectively. Prove that 
.
be a triangle with orthocenter and midpoints , and of , , and respectively. A circle with center intersects the lines , , and in points respectively. Prove that .
is 
. Because of 
midpoint s we have parallel to so is perpendicular bisector of altitude from .
so we have 
. Then we have 
. If we do pisagor theorem we have 
so 
. And 
.
. let 
. So we have 
. Similarly we have 
, 
, 
. So we have and we getresult....
:
Where 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
