B4. Emile stands in a circle with nine other people. Each of the ten people thinks of an integer (which can also be negative) and whispers this number to both of their neighbors. Then, each person loudly states the average of the two numbers they heard from their neighbors. It turns out that Emile says the number 10, his right neighbor says the number 9, the next right neighbor says the number 8, and so on around the circle until Emile's left neighbor, who says the number 1.

What number did Emile have in mind?
B4. 5

The number that Emile had in mind we call $c_{10}$, the number of his right neighbor we call $c_{9}$, and so on until the left neighbor of Emile, who had the number $c_{1}$ in mind. We then see that

\[
\begin{aligned}
c_{10}+c_{8} & =2 \cdot 9=18, \\
c_{8}+c_{6} & =2 \cdot 7=14, \\
c_{6}+c_{4} & =2 \cdot 5=10, \\
c_{4}+c_{2} & =2 \cdot 3=6, \\
c_{2}+c_{10} & =2 \cdot 1=2 .
\end{aligned}
\]

Adding gives $2\left(c_{2}+c_{4}+c_{6}+c_{8}+c_{10}\right)=50$, so $c_{2}+c_{4}+c_{6}+c_{8}+c_{10}=25$.

It follows that $c_{10}=\left(c_{2}+c_{4}+c_{6}+c_{8}+c_{10}\right)-\left(c_{2}+c_{4}\right)-\left(c_{6}+c_{8}\right)=25-6-14=5$.