Exercise 2. Each strictly positive rational number is colored either red or black, such that:

- the numbers $x$ and $x+1$ are of different colors;
- the numbers $x$ and $\frac{1}{x}$ are of the same color;
- the number 1 is colored red.

What is the color of $\frac{2012}{2013}$?

(No proof of the existence of such a coloring is required.)
## Solution.

We notice that $\frac{2012}{2013}$ can be written as:

$$
\frac{2012}{2013}=\frac{1}{1+\frac{1}{2012}}
$$

Thus, according to the rules of the problem, it has the opposite color of 2012. Now, the latter is obtained by adding 1 to 1, 2011 times; it therefore has the opposite color of 1 (since the color is inverted 2011 times), which is black. Thus, $\frac{2012}{2013}$ is red.

![](https://cdn.mathpix.com/cropped/2024_06_04_6c96fb1ce2d0d58c89f6g-03.jpg?height=89&width=187&top_left_y=969&top_left_x=969)