AoPS Academy
and 
be positive real numbers with 
. Prove that
For which and equality holds?
, find the number of terms with coefficient divisible by 
.
and 
of an equilateral triangle 
are taken points 
and 
, and on the side 
, points 
and 
so that 
. Prove that the angle between lines 
and 
is equal to 
.
over 
. Ignore "pathological" solutions.[/quote]
![\[\lfloor x \rfloor\]](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 .
Is there an elegant way to prove this?
. They are drawn inside a square of area 
. Find the greatest value of 
such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than .
. They are drawn inside a square of area . Find the greatest value of such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than .
such that 
for all 
and 
Prove that
Prove that
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](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.
