[/quote]
,
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)
that satisfy the equation
with incenter 
.
and 
be points on the side BC such that 
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 reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
the intersection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
is an equilateral . Let 
and 
at 
and 
at 
Y by
Let 
be the midpoint of arc 
not containing 
on the circumcircle of a given triangle 
. Let 
be the incenter of , and let 
and 
be the projections of onto 
and 
, respectively. Prove that 
.
be the midpoint of arc 
not containing 
on the circumcircle of a given triangle 
. Let 
be the incenter of , and let 
and 
be the projections of onto 
and 
, respectively. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Oct 29, 2020, 8:23 PM
to get that:
since we have that 
we have that:
and 
.
and 
,
and that 
holds
we have that 
,
.
