with incenter 
.
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)
we have 
.
.
be reals such that 
and 
Prove that
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
, points 
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)
of the pizza, where 
is a positive integer. Find 
Y by DPS, Adventure10, Mango247
A semicircle is given with diameter 
. On arc of the semicircle, an arbitrary point 
is chosen that differs from points 
and 
. Let 
be orthogonal projection of point on the diameter . A circle \omega touches segments 
and arc at point 
. Prove that the intersection point of bisectors of angles 
and 
lies on the diameter .
(V. Yasinskyy, Vinnytsya)
. On arc of the semicircle, an arbitrary point 
is chosen that differs from points 
and 
. Let 
be orthogonal projection of point on the diameter . A circle \omega touches segments 
and arc at point 
. Prove that the intersection point of bisectors of angles 
and 
lies on the diameter .(V. Yasinskyy, Vinnytsya)
with 
be 
.
bisects 
.
.
is the bisector of 
