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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?
be a triangle with 
and 
. Let be the midpoint of . is the point on 
such that 
. Let 
be any point distinct from . Prove that 
.
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.
and 
So we see that in this case 
Additionally since 
we know that 
so 
From here we prove that 
where x z and y are all positive. From here is algebra ![\[3x+z<2\sqrt{3x^2+z^2}\]](http://latex.artofproblemsolving.com/2/1/6/2166c337ca41871aaeca16c526242b353ebe4a5b.png)
![\[9x^2+6xz+z^2<12x^2+4z^2\]](http://latex.artofproblemsolving.com/c/2/0/c20a8dae42455ced1e8ca3be7aba527a91936321.png)
we are allowed to square without changing sign as we are dealing with positives. ![\[6xz<3x^2+3z^2\]](http://latex.artofproblemsolving.com/b/8/2/b82723524e1f94ea4765c6eb7956e48519ff3c4d.png)
![\[2xz<x^2+z^2\]](http://latex.artofproblemsolving.com/3/b/e/3be557ba1302786036fd8d8c1a0067f9a69180f4.png)
![\[\sqrt{xy}<\sqrt{\frac{x^2+z^2}{2}}\]](http://latex.artofproblemsolving.com/1/f/6/1f68f03be2312b71dab244fe6251418fb79c5083.png)