be the set of positive real numbers. Find all functions 
such that, for all 
,
,
be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of 
-
as many lattice points as does 
,
of the lattice points contained in 
is 27 times the fraction of lattice points in 
.
and 
be positive real numbers with 
.
be the maximum possible value of 
for which the system of equations![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](http://latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
.
can be expressed as a fraction 
,
and 
are relatively prime positive integers. Find 
.
through 
.
.
,
,
with probability 
and to pad 
with probability 
.
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 
,
