and 
Prove that
![\[\lfloor x \rfloor\]](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)
Prove that
of the pizza, where 
is a positive integer. Find 
Y by
In triangle 
, points 
and 
are the intersection points of a straight line parallel to 
, passing through the vertex 
, with bisectors of the external angles 
and 
of the triangle, respectively. The perpendicular on the line 
at point , and the perpendicular on the line 
at point , intersect at the point 
. Let 
be the incenter of the triangle . Prove that 
.
, points 
and 
are the intersection points of a straight line parallel to 
, passing through the vertex 
, with bisectors of the external angles 
and 
of the triangle, respectively. The perpendicular on the line 
at point , and the perpendicular on the line 
at point , intersect at the point 
. Let 
be the incenter of the triangle . Prove that 
.![[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair Ia = dir(120), Ib = dir(210), Ic = dir(330), A = foot(Ia,Ib,Ic), B = foot(Ib,Ic,Ia), C = foot(Ic,Ia,Ib), I = extension(Ia,A,Ib,B), R = 2A-I, P = foot(R,Ic,Ia), Q = foot(R,Ia,Ib);
dot("$I_A$", Ia, dir(120));
dot("$I_B$", Ib, dir(180));
dot("$I_C$", Ic, dir(330));
dot("$A$", A, dir(135));
dot("$B$", B, dir(45));
dot("$C$", C, dir(150));
dot("$I$", I, dir(45));
dot("$P$", P, dir(60));
dot("$Q$", Q, dir(200));
dot("$R$", R, dir(270));
draw(Ia--Ib--Ic--cycle, linewidth(1.1));
draw(unitcircle^^circumcircle(A,R,P)^^circumcircle(A,R,Q), gray);
draw(Ib--B--C--Ic, dotted);
draw(Ib--Q--R--P--Q^^Ia--R);
[/asy]](http://latex.artofproblemsolving.com/4/b/7/4b796b2f9b73d0bc1328a6e6b21187f48ed103ac.png)
be the 
be the 
be the 
and 
is cyclic because 
.
so 
and it is the reflection of 
as desired.
