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, Mango247, NicoN9
Suppose that 
boys and 
girls line up in a row. Let 
be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row 
we have 
. The average value of (if all possible orders of the 20 people are considered) is closest to
boys and 
girls line up in a row. Let 
be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row 
we have 
. The average value of (if all possible orders of the 20 people are considered) is closest to
.
or 
.
), we find that the total number of ways to arrange the remaining 18 people is 
.
.
.
,
possible combinations for the first two positions, of which 
have a boy and girl next to each other. Thus, the probability that the first two people in the row are a boy and a girl is 
Thus, the expected value of 
The answer is therefore 
.
,
.
,
.
,
.
chance of having boy and girl
,
,
,![\[
X_i =\begin{cases}
1 \text{ if $P_i$ and $P_{i + 1}$ are different gender } \\
0 \text{ else. }
\end{cases}
\]](http://latex.artofproblemsolving.com/5/4/b/54b9ed4e1c5af2b2e19ff4a1b05eddce043b2099.png)
We want 
,![\[
\mathbb{E}(X_1+\dots +X_{19}) =19\cdot \mathbb{E}(X_1) =\frac{91}{10}.
\]](http://latex.artofproblemsolving.com/f/f/d/ffdfeda96817c173c2d096a55cc4f320a0977b2c.png)
(Here we used 
,
