and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
of an equilateral triangle 
are taken points 
and 
,
, 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 ProceraDEU235, RedFireTruck
Thia is is a 
dimensional problem, i.e. it is a problem in the plane. The effective length 
of a pipe is the distance between the end points. Find the rigid pipe of longest effective length that can be taken around the corner in the above corridor. Is there a unique shape such a piece of pipe must have?
dimensional problem, i.e. it is a problem in the plane. The effective length 
of a pipe is the distance between the end points. Find the rigid pipe of longest effective length that can be taken around the corner in the above corridor. Is there a unique shape such a piece of pipe must have?This post has been edited 1 time. Last edited by parmenides51, Jan 27, 2021, 11:19 PM
