that satisfy the equation
.
at 
.
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)
'is a perfect square.
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
so that 
.
and 
is equal to 
.
over 
.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Y by Adventure10
Let 
and 
be given. Let 
be a triangle with 
and 
, and call the cicumcentre 
, and the orthocentre 
. The point 
lies on the ray from 
through . Let 
be the mirror image of through 
, and 
the mirror image of through 
. Assume that 
is cyclic. Show that the length 
depends only on 
and 
.
and 
be given. Let 
be a triangle with 
and 
, and call the cicumcentre 
, and the orthocentre 
. The point 
lies on the ray from 
through . Let 
be the mirror image of through 
, and 
the mirror image of through 
. Assume that 
is cyclic. Show that the length 
depends only on 
and 
.
