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 
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: 
the points and are marked on the sides 
and respectively, in such a way that 
and 
. Let be the intersection between 
and 
. If the side of the square equals 
, what is the distance from point to side ?
the points and are marked on the sides and respectively, in such a way that and . Let be the intersection between and . If the side of the square equals , what is the distance from point to side ?
, then 
.
cuts the line 
in the point 
.
.
Prove that
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
and 
.
Let 
.
and 
Prove that
so 
with a similarity ratio of 