[/quote]
,
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
with incenter 
.
at 
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
is the reflection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
Y by Adventure10, Mango247
Two radii 
and 
of a unit circle form an angle 
, 
. Let 
be an arbitrary point on the arc 
. A ray of light from is reflected from the segments 
and the arc so that it moves along the sides of a fixed triangle 
, with 
on and R on . Prove that the perimeter of 
does not depend on , and find it.
and 
of a unit circle form an angle 
, 
. Let 
be an arbitrary point on the arc 
. A ray of light from is reflected from the segments 
and the arc so that it moves along the sides of a fixed triangle 
, with 
on and R on . Prove that the perimeter of 
does not depend on , and find it.
