and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
,
, points 
and 
so that 
.
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
Inside a circle with radius 
lie four smaller circles with centres 
and 
. The circles touch each other as shown. The point where the circles with centres 
and 
touch each other is the centre of the big circle. Calculate the area of quadrilateral 
.
lie four smaller circles with centres 
and 
. The circles touch each other as shown. The point where the circles with centres 
and 
touch each other is the centre of the big circle. Calculate the area of quadrilateral 
.
.
intersect at right angles.
and 
.
,
by Pythagorean Theorem. We can rewrite this equation as ![\[(3+r)^2=3^2+(6-r)^2,\]](http://latex.artofproblemsolving.com/6/a/e/6ae0c110b8026a3c4630a83785aae7deeb8299b5.png)
which simplifies to![\[9+6r+r^2=9+36-12r+r^2,\]](http://latex.artofproblemsolving.com/4/f/3/4f362f1aa55a68da49b4a7cfa65d7bb459a291c6.png)
![\[18r=36,\]](http://latex.artofproblemsolving.com/d/d/4/dd4b635aab7d094d95042807c548523a057eba9a.png)
![\[r=2.\]](http://latex.artofproblemsolving.com/2/4/d/24d0cda8f3f7be151d742d623501bfab44d8109f.png)
,
,
is a 
right triangle. Hence, the diagonals of rhombus 
,
.
