and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
of an equilateral triangle 
are taken points 
,
, points 
.
and 
is equal to 
.
over 
.
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 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Is there an elegant way to prove this?
Y by Adventure10
Consider a semicircle with diameter 
, center 
and radius 
. Let 
be the point of segment such that 
. Line 
is perpendicular to on and 
is the intersection point of and the semicircle. Let 
be the foot of the perpendicular drawn from on 
and 
the point of intersection of lines 
and 
.
a) Calculate in terms of .
b) If
and 
are the midpoints of 
and 
respectively, find the measure of the angle 
.
, center 
and radius 
. Let 
be the point of segment such that 
. Line 
is perpendicular to on and 
is the intersection point of and the semicircle. Let 
be the foot of the perpendicular drawn from on 
and 
the point of intersection of lines 
and 
.a) Calculate
in terms of .b) If
and 
are the midpoints of 
and 
respectively, find the measure of the angle 
.
